Finite Volumesmes
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Finite Volumesmes

for Complex Applications II

Sponsored by :

© HERMES Science Publications, Paris, 1999 HERMES Science Publications 8, quai du Marche-Neuf 75004 Paris Serveur web : http://www.hermes-science.com ISBN 2-7462-0057-0 Catalogage Electre-Bibliographie Finite Volumes for Complex Applications II — Problems and Perspectives Vilsmeier, Roland* Benkhaldoun, Fayssal* Hanel, Dieter Paris : Hermes Science Publications, 1999 ISBN 2-7462-0057-0 RAMEAU : elements finis, methode des analyse numerique DEWEY : 515 : Analyse mathematique Le Code de la propriete intellectuelle n'autorisant, aux termes de 1'article L. 122-5, d'une part, que les « copies ou reproductions strictement reservees a 1'usage prive du copiste et non destinees a une utilisation collective » et, d'autre part, que les analyses et les courtes citations dans un but d'exemple et d'illustration, « toute representation ou reproduction integrate, ou partielle, faite sans le consentement de 1'auteur ou de ses ayants droit ou ayants cause, est illicite » (article L. 122-4). Cette representation ou reproduction, par quelque precede que ce soit, constituerait done une contrefacon sanctionnee par les articles L. 335-2 et suivants du Code de la propriete intellectuelle.

Finite Volumes for Complex Applications II Problems and Perspectives

editors Roland Vilsmeier Fayssal Benkhaldoun Dieter Hanel

Second International Symposium on Finites Volumes for Complex Applications Problems and Perspectives July 19-22, 1999, Duisburg, Germany Internet adress : http://www.vug.uni-duisburg.de/FVCAII/contents.html

Organizing Institutions Institut fur Verbrennung und Gasdynamik (IVG), University Duisburg, Germany INSA de Rouen, France

Scientific Committee F. Benkhaldoun, LMI, INSA de Rouen, France R. Borghi ESM2, IMT-Technopole, Marseille, France A. Dervieux, INRIA Sophia Antipolis, France T. Gallouet, Universite Aix-Marseille I, France D. Hanel, IVG, University Duisburg, Germany D. Kroner, Institute f. Angewandte Mathematik, University Freiburg, Germany I. Toumi, CEA, Saclay, France J.-R Vila, INSA de Toulouse, France R. Vilsmeier, IVG, University Duisburg, Germany N. R Weatherill, University of Swansea, UK G. Wittum, IWR, University Heidelberg, Germany

Invited Keynote Lectures R. Abgrall, Universite de Bordeaux 1, France F. Coquel, CNRS, Paris, France G. Degrez, von Karman Institute, St-Genesius-Rode, Belgium R. Klein, Konrad-Zuse-Zentrum f. Informationstechnik, Berlin, Germany R. Lazarov, Lawrence Livermore National Laboratory, USA J. M. Ghidaglia, ENS de Cachan, France S. Noelle, Institute f. Angew. Mathematik, University Bonn, Germany

Contents

Editors preface

XIII

Invited speakers

1

Construction of some genuinely multidimensional upwind distributive schemes — R. ABGRALL

3

A Roe-type Linearization for the Euler Equations for Weakly Ionized Gases

F. COQUEL, C. MARMIGNON

11

Multidimensional Upwind Residual Distribution Schemes and Application

H. DECONINCK, G. DEGREZ

27

Overcoming mass losses in Level Set-based interface tracking schemes

Th. SCHNEIDER, R. KLEIN

41

Coupling mixed and finite volume discretizations of convection-diffusionreaction equations on non-matching grids R.D. LAZAROV, J.E. PASCIAK, P.S. VASSILEVSKI

51

Numerical computation of 3D two phase flow by finite volumes methods using flux schemes — J. M. GHIDAGL1A

69

The MoT-ICE : a new high-resolution wave-propagation algorithm based on Fey's Method of Transport — S. NOELLE

95

Numerical Analysis

115

Error estimate for a finite volume scheme on a MAC mesh for the Stokes problem — P. BLANC

117

Convergence Rate of the Finite Volume Time-explicit Upwind Schemes for the Maxwell System on a Bounded domain Y. COUDIERE, P. VlLLEDEU

125

VI

Finite volumes for complex applications

Flux vector splitting and stationary contact discontinuity — F. DUBOIS

133

Analysis of a Finite Volume Solver for Maxwell's Equations F. EDELVIK

141

A result of convergence and error estimate of an approximate gradient for elliptic problems — R. EYMARD, T. GALLOUET, R. HERBIN

149

Finite volume approximation of elliptic problems with irregular data T. GALLOUET, R. HERBIN

155

Analysis of a finite volume scheme for reactive fluid flow problems A. HOLSTAD, I. LIE

163

Convergence of a finite volume scheme for a nonlinear convectiondiffusion problem — A. MICHEL

173

Convergence analysis of a cell-centered FVM H.P. SCHEFFLER, R. VANSELOW

181

Error estimates on the approximate finite volume solution of convection diffusion equations with boundary conditions T. GALLOUET, R. HERBIN, M. H. VIGNAL

189

The limited analysis in finite elasticity — LA. BRIGADNOV

197

Entropy consistent finite volume schemes for the thin film equation G. GRUN, M. RUMPF

205

Convergence of finite volume methods on general meshes for non smooth solution of elliptic problems with cracks P. ANGOT, T. GALLOUET, R. HERBIN

215

Application and analysis of finite volume upwind stabilizations for the steady-state incompressible Navier-Stokes equations — L. ANGERMANN

223

A new cement to glue non-conforming grids with Robin interface conditions Y. ACHDOU, C. JAPHET, F. NATAF, Y. MADAY

231

Finite Volume Box Schemes — J.-P. CROISILLE

239

On nonlinear stability analysis for finite volume schemes, plane wave instability and carbuncle phenomena explanation — M. ABOUZIAROV

247

Innovative schemes

253

A comparison between upwind and multidimensional upwind schemes for unsteady flow — P. BRUFAU, P. GARCIA-NAVARRO

255

Reformulation of the unstructured staggered mesh method as a classic finite volume method — B. PEROT, X. ZHANG

263

A mixed FE-FV algorithm in non-linear solid dynamics — S.V. POTAPOV

271

Contents

VII

An Euler Code that can compute Potential Flow — M. RAD, P. ROE

279

Finite volume evolution Galerkin methods for multidimensional hyberbolic problems — M. LUKACOVA-MEDVIDOVA, K.W. MORTON, G. WARNECKE

289

Nonlinear anisotropic artificial dissipation - Characteristic filters for computation of the Euler equations T. GRAHS, A. MEISTER, T. SONAR

297

Nonlinear projection methods for multi-entropies Navier-Stokes systems C. BERTHON, F. COQUEL

307

About a Parallel Multidimensional Upwind Solver for LES D. CARAENI, S. CONWAY, L. FUCHS

315

A higher-order-accurate upwind method for 2D compressible flows on cell-vertex unstructured grids — L. CATALANO

323

A New Upwind Least Squares Finite Difference Scheme (LSFD-U) for Euler Equations of Gas Dynamics — N. BALAKRISHNAN, C. PRAVEEN

331

A finite-volume algorithm for all speed flows F. MOUKALLED, M. DARWISH

339

Preserving Vorticity in Finite-Volume Schemes — P. ROE, B. MORTON

347

On Uniformly Accurate Upwinding for Hyperbolic Systems with Relaxation — J. HITTINGER, P. ROE

357

Implicit Finite Volume approximation of incompressible multi-phase flows using an original One Cell Local Multigrid method S. VINCENT, J.P. CALTAGIRONE

367

New classes of Integration Formulas for CVFEM Discretization of ConvectionDiffusion Problems — E. P. SHURINA, T.V. VOITOVICH

377

Fields of application

385

Analysis of Finite Volume Schemes for Two-Phase Flow in Porous Media on Unstructured Grids — M. AFIF, B. AMAZIANE

387

A preconditioned finite volume scheme for the simulation of equilibrium two-phase flows — S. CLERC

395

Transient flows in natural valleys computed on topography-adapted mesh S. SOARES FRAZAO, J. LAU MAN WAI, Y. ZECH

403

A mixed Finite Volume/Finite Element method applied to combustion in multiphase medium — N. GUNSKY-OLIVIER, E. SCHALL

411

Turbulence Modeling for Separated Flows — L.J. LENKE, H. SMON

419

VIII

Finite volumes for complex applications

Simulation of unsteady Flow in a Vortex-Shedding Flowmeter S. PERPEET, A. ZACHCIAL, E. VON LAV ANTE

429

A Finite Volume Scheme for the Two-Scale Mathematical Modelling of TiC Ignition Process — A. AouFi, V. ROSENBAND

437

Two Perturbation Methods to Upwind the Jacobian Matrix of Two-Fluid Flow Models — A. KUMBARO, I. TOUMI, J. CORTES

445

Finite volumes simulations in magnetohydrodynamics — M. HUGHES, L. LEBOUCHER, V. BOJAREVICS, K. PERICLEOUS, M. CROSS

453

Finite Volume Method for Large Deformation with Linear Hypoelastic Materials — K. MANEERATANA, A. IVANKOVIC

459

A finite volume formulation for fluid-structure interaction C.J GREENSHIELDS, H.G. WELLER, A. IVANKOVIC

467

Boundary Conditions for Suspended Sediment V. BOVOLIN, L. TAGLIALATELA

475

Second order corrections to the finite volume upwind scheme for the 2D Maxwell equations — B. BlDEGARAY, J.-M. GHIDAGLIA

483

A MHD-Simulation in the Solar Physics A. DEDNER, C. ROHDE, M. WESENBERG

491

A Zooming Technique for Wind Transport of Air Pollution P.J.F. BERKVENS, M.A. BOTCHEV, W.M. LIOEN, J.G. VERWER

499

Computational Solid Mechanics using a Vertex-based Finite Volume Method G.A. TAYLOR, C. BAILEY, M. CROSS

507

Control volumes technique applied to gas dynamical problems in underground mines — E. VLASSEVA

517

Simulation of salt-fresh water interface in costal aquifers using a finite volume scheme on unstructured meshes — B. BOUZOUF, D. OUAZAR, I. ELMAHI

525

Progress in the flow simulation of high voltage circuit breakers X. YE, L. MULLER, K. KALTENEGGER, J. STECHBARTH

533

River valley flooding simulation — F. ALCRUDO

543

Modelling vehicular traffic flow on networks using macroscopic models

J.P. LEBACQUE, M.M. KHOSHYARAN

551

Finite Volume method applied to a solid/liquid phase change problem M. ELGANAOUI, P. BONTOUX, O. MAZHOROVA

559

Integrating finite volume based structural analysis procedures with CFD software to analyse fluid structure interactions M.A. WHEEL, A. OLDROYD, T.J. SCANLON, P. WENKE

567

Contents

IX

A generalized parcel method for the spray dispersion computation

B.NKONGA

575

Finite Volume Methods for Multiphysics Problems — C. BAILEY, M. CROSS, K. PERICLEOUS, G.A. TAYLOR, N. CROFT, D. WHEELER, H. Lu

585

A Finite Volume discretization and multigrid solver for steady viscoelastic fluid flows — H. AL MOATASSIME, S. RAGHAY, A. HAKIM

595

Complexity, Performance and Informatics

605

Various CG-type methods applied to finite volume schemes O. SCHMID, A. BUBMANN, E. VON LAV ANTE, M. MOCZALA

607

A Newton-Relaxation Finite Volume Scheme for Simulation of Dynamic Motion — B.A. JOLLY, M. RlZK

615

An Attempt to Develop a Multi Purpose FAS Multigrid Algorithm L. FOURNIER, O. GLOTH

623

On Higher Order Accurate Implicit Time Advancing for Stiff Flow Problems C. VlOZAT, E. SCHALL, A. DERVffiUX, D. LESERVOISIER

631

Numerical Solution of Steady 2D and 3D Impinging Jet Flows K. KOZEL, P. LOUDA, J. PRIHODA

639

Triangular, Dual and Barycentric Finite Volumes in Fluid Dynamics J. FELCMAN, M. FEISTAUER

647

Concepts for parallel numerical solution of PDEs — G. BERTI

655

Performing parallel direct numerical simulation of two dimensional heated jets S. BENAZZOUZ, V.G. CHAPIN, P. CHASSAING

663

Two-Dimensional Riemann Problems. Assessment Tests for Upwind Methods for Multi-Dimensional Supersonic Flow Problems J. VAN KEUK, J. BALLMANN

671

Robustness and accuracy on unstructured grids. Numerical experiments on finite volume schemes — E.A. MEESE, S.E. HAALAND

683

A validation of an efficient numerical method for 3-D complex flows

E.A. FADLUN, S. LEONARDI, R. VERZICCO, P. ORLANDI

693

Comparison of Two Finite Volume Methods for 3D Transonic Flows through Axial Cascades — J. FORT, J. FORST, J. HALAMA, K. KOZEL

701

An efficient and universal numerical treatment of source terms in turbulence modelling — B. MERCI, J. STEELANT, E. DICK

709

Comparison of numerical solvers for a multicomponent, turbulent flow E. XEUXET, A. FORESTIER, J.M. HERARD

717

X

Finite volumes for complex applications

Parallel Overlapping Mesh Technique for Compressible Flows J. ROKICKI, D. DRIKAKIS, J. MAJEWSKI, J. ZOLTAK

725

A comparison of Finite Volume and Higher-Order Finite Difference Schemes for the Solution of the Navier-Stokes and Euler equations M. MEINKE, Th. RISTER, R. EWERT

733

Simulation of 3D turbulent flow through steam-turbine control valves B.N. AGAPHONOV, V.D. GORYACHEV, V.G. KOLYVANOV, V.V. Ris, E.M. SMIRNOV, D.K. ZAITSEV

743

Adaptivity, Tracking and Fitting

751

An Adaptive Hybrid Object-Oriented Code for CFD-Applications-Adhoc3D U. TREMEL, H. BLEECKE, G. BRENNER, G. GREINER

753

Adaptive mesh refinement for single and two phase flow problems in porous media — M. OHLBERGER

761

Parallel solution of hyperbolic PDEs with space-time adaptivity P. LOTSTEDT, S. SODERBERG

769

Dynamic mesh generation with grid quality preserving methods

A. WICK, F. THELE A Finite Volume Method for Steady Hyperbolic Equations M. J. BAINES, SJ. LEARY, M.E. HUBBARD

777 787

Moving grid technology for finite volume methods in gas dynamics B.N. AZARENOK, S.A. IVANENKO

795

Numerical Simulation of Lifted Turbulent Methane-Air Diffusion Flames M. CHEN, N. PETERS

803

The application of a conservative grid adaption technique to 1D unsteady problems — M. CASTRO-DIAZ, P. GARCIA-NAVARRO

809

Application of mesh adaptive techniques to mesh convergence in complex CFD D. LESERVOISIER, A. DERVIEUX, P.L. GEORGE, O. PENANHOAT

817

Multidimensional Fully Adaptive Finite Volume Schemes for the Numerical Simulation of Stiff Combustion Front Propagation in Condensed Phase A. AOUFI

825

Mathematical and numerical modeling of a two-phase flow by a Level Set method — S. ROUY, P. HELLUY

833

Multiresolution analysis on triangles: application to conservation laws

A. COHEN, S.M. KABER, M. POSTEL A local level set method for the treatment of discontinuities on unstructured grids — L. TRAN, R. VILSMEIER, D. HANEL

841 849

Contents

XI

Varia

857

A Stabilized Version of Wang's Partitioning Algorithm for Banded Linear Systems — V. PAVLOV

859

On Jeffreys Model of heat conduction — M. DRYJA, K. MOSZYNSKI

867

Investigation of some method for cavitating jet S. OCHERETYANY, V.V. PROKOFIEV

Index des auteurs

875

887

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Editors preface Finite Volume methods are methods directly related to the numerical solution of conservation laws. Systems of such conservation laws govern wide fields of physics and the efficiency of corresponding solution methods is an essential requirement from basic research and industry. Since the efficiency of any method must be measured by the quality of the result compared to the computational cost to spend for, corresponding developments are widely spread, ranging from very fundamental numerical analysis up to the efficient use of modern computer hardware. Although in the past the numerical methodology has made large progresses, many problems and difficulties remain, requirering further intensive research. New fields of application as well as the coupled simulation of different physical phenomena become accessible due to improved solution techniques and growing computer capacities. However, these new possibilities introduce again new physical and mathematical problems to be solved. The development of new methods as well as the extension of existing ones requires intensive and critical investigations and careful validation. One of the aims of this conference is therefore to bring together people working in theory and practice for fruitful and critical discussions about methods, their advantages and drawbacks and related experiences from arbitrary applications. The present proceedings summarize the contributions to be presented at the second international symposium on Finite Volumes for Complex Applications Problems and Perspectives. The first symposium of this series was held summer 1996 at INSA de Rouen in France. Based on the success of this first conference, the symposium in Duisburg has again received an unexpected high attention in the numeric community. After a critical review of the submitted contributions, 98 papers by authors from 20 countries are presented in this volume. In a rough estimation, about half of the contributions can be assigned to analysis and numerics of different methods whereas the other half is essentially concerned with application and computational aspects of methods. We would like to thank all persons, who contributed to the conference and to this book of proceedings. First of all, we want to mention all the authors as well as the other members of the scientific committee, for the work of writing the papers as well as selecting these in remarkably short times. We would like to extend our thanks, acknowledging the help from the numerical staff, secretaries and students at the IVG in Duisburg and at INSA de Rouen, keeping their good mood whenever. Finally we want to thank the following organizations for the financial support: Deutsche Forschungsgemeinschaft, Ministerium fur Wissenschaft und Forschung Nordrhein-Westfalen, Duisburger Universitatsgesellschaft, Hewlett Packard and Sun Microsystems.

Fayssal Benkhaldoun, Dieter Hanel and Roland Vilsmeier

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Invited speakers

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Construction of genuinely multidimensional upwind distributive schemes

Remi Abgrall Universite Bordeaux I Mathematiques Appliquees 351 Cours de la Liberation 33 405 Talence Cedex, France

ABSTRACT We present a construction of a class of upwind residual schemes for the Euler equations of fluid mechanics. It naturaly generalises the PSI scheme of Deconinck, Roe and Struijs. We show some of its theoretical properties. Numerical illustrations indicates obvious advantages over the now classical finite volumes upwind schemes. Key Words : Upwind schemes, Residual schemes, Unstructured meshes.

1. Introduction Most of the "modern" industrial codes that are used to simulate complex compressible fluids are written according to the ideas developed in the 80's by van Leer, Harten, Osher, Roe, and many others. These codes are versatile, robust and rather accurate. However, in some situations, the results are somewhat disapointing : the numerical viscosity of these schemes is still too high, despites some attemps to reduce it. It is still difficult to accurately compute the lift and drag of an airfoil. These schemes are also very mesh dependent : the formulation deeply rely on the shape of the control volume. Their definition is many times not related to the physics of the problem under consideration. For these reasons, since a few years, some researchers have tried to find alternative formulations of the problem that could - in principle - lead to less dissipative and mesh independant schemes. Among the first, one might quote [Da] who tried to represent the fluxes in term of Riemann problems in direction related to the fluid. A very promissing formulation is certainly that of upwind residual schemes, who were pioniered by P.L. Roe, the H. Deconinck and their coau-

4

Finite volumes for complex applications

thors. These schemes are also related to the SUPG schemes of Hughes, or the streamline diffusion method by Johnson. If the design principle of the upwind residual schemes are rather clear for scalar convection equations, this is not anymore true for systems, and in particular the Euler equations of fluid mechanics. The aim of the present paper is to give the status of a research toward that goal. The paper is divided into four parts. In the first one, we recall the upwind residual formulation and the design principle for scalar equations. Then we reformulate the PSI scheme of Deconinck, Roe, Struijs and Sidilkover in a way that is easier to generalise in multiD. More precisely, it is seen as a blending between the N scheme and the LDA scheme. In the third part, we discuss design principle for the Euler equations, and propose a class of schemes. Then numerical examples are given. 2. Upwind Residual schemes 2.1 The Euler equations We consider the Euler equations for fluid mechanics with initial and boundary conditions

inflow and/or wall conditions. In (1), the vector of conserved variables is W — (p, pu, pv, E)T, the x-component of the flux is Fx = (pu,pu2 + p, puv, u(E + p))T. Its y-component has a similar expression. In the problem of interest, the pressure p is related to W via p = (7 _ ! ) ( £ _ _ p ( u 2 + v 2 )) with 7 = 1.4. The solutions of (1) has to fullfill £ the second law of thermodynamics : we have to have

where S = —ps (with s = log ( -^-)) is the mathematical entropy. In [Ta], E. Tadmor has shown that the solutions of (1-2) satisfy the following minimum principle

Invited speakers

5

2.2 Numerical schemes

2.2.1 Generalities The discretisation of (1) is carried out on a mesh made of triangles. The list of nodes is {Mi} i = 1 , n s . The generic name of a triangle is T, its vertices are denoted by Mi1; Mi2, Mi3, or 1, 2, 3 when there is no ambiguity. The schemes for (1) are written as

In (4), W™ is an approximation for W(Mi,nA£), \Ci\ is the area of the dual cell associated to node Mi1 The residual $iT must satisfy

In this conservation relation Fh is an approximation of F that has to be continuous across the edges of the triangles. Under classical asssumptions, we have a Lax Wendroff-like theorem [AMN] : the scheme, if it converges, converges to weak solutions of (1). In this paper, we follow the approach of Roe-DeconinckStruijs [SDR] via the parameter vector Z — (^/p, <

where 2Ki = ( n i ) x A + (ni) y B. One can show that Ki is diagonalisable and has real eigenvalues.

2.2.2 Design principles The schemes are constructed follwing three design principles - the scheme is upwind : if Ki only has negative eigenvalues, $J = 0. ^n particular, we have |Ci| = \ ^T M - P T l^l-

6

Finite volumes for complex applications

- the scheme must be linear preserving : if $T = 0 then

In the definition of these schemes, the matrix N appears. It is the inverse of X]i=i ^"i~- This matrix is not always invertible. However, for the Euler equations, one can show [AMN] that it is always invertible except at stagnation points. In any case, one can always give a meaning to K^Z or N$T because the Euler equations are symetrizable, see [AMN]. Hence, there is no problem in the definition of ^f7 or $fDA. Both schemes are clearly upwind. The LDA scheme is linear preserving contrarily to the N scheme. The N scheme is monotone. This is very obvious for its scalar version because in that case we have

Invited speakers

7

with Cij > 0. The matrix equivalent formulation involve terms like K^NK~ that are difficult to handle. In all numerical experiments (with a large variety of geometries and flow conditions) seem to indicate that the system N scheme also satisfies a discrete version of (3). Barth [Ba] has shown that the N scheme, for a linear symetric hyperbolic system is locally dissipative. In [Ab], we show that the LDA scheme is also locally dissipative for a linear symetric hyperbolic system. 3. A positive linear preserving scheme for scalar equation : the PSI scheme revisited We consider the scalar versions of the N and LDA schemes and the following blended scheme The firt remark is that this scheme satisfies the conservation constraint (5) whatever / e R. This scheme is upwind by construction since the N and LDA schemes are upwind. We now consider the positivity issue with the same tech/

nique as D. Sidilkover, $< = l^

&LDA\

+ (l-l)^DA = (/ + (!- O-^T ) • Thanks V

^i

/ If we set

the positivity is obtained if

a solution is given by

where 0 and

T

else. Simple algebraic manipulations r —I shows that if / is chosen with the = sign in (7), the scheme is positive and linear preserving. In fact, it is identical to the PSI scheme. 4. A scheme for the Euler equations Following the same ideas, we consider a scheme written as

where 1 is a matrix. In order to illustrate the design principles, we consider 1 = 1 = Id where / 6 M, but a more sophisticated method is developped in [Ab].

8

Finite volumes for complex applications Conditions Top Bottom

P 1.4 0.7

P 1 0.25

Mach number 2.4 4

TAB. 1: Conditions for the interaction of 2 parallel supersonic flows Let us denote by

and thanks to the numerical experiments, we assume that the system N scheme has a local minimum principle for the specific entropy. Following the same arguments as for the scalar PSI scheme, the conditions are

where here

One can show that this scheme is also locally dissi-

pative[Ab]. 5. Numerical experiments

We present some results obtained in two different test cases. The first one is the iteraction of two parallel supersonic flows. The conditions are given in Table 1 The mesh is given on Figure 1-a The isovalues of the density are presented on Figure 1-b. A very clear improvement of the results can be observed. The new scheme give monotone results that are more accurate than those of the finite volume scheme (MUSCL extrapolation on conserved variables). The second test case is a GAMM test case : Naca0012, Mach number : 0.85, angle of incidence : 1 degree. We show the isolines of the Mach number (Figure 2-a) and the isoline of the reduced entropy (Figure 2-b). It is clear that the slip line out of the leading edge is improved as well as the entropy profiles. 6. Conclusions

We have sketched the construction of upwind residual schemes that are also linear preserving. Some numerical example indicate that these new schemes are more accurate than the now classical finite volume schemes.

Invited speakers

9

Bibliography

[Ab]

R. ABGRALL. Upwind residual schemes on unstructured meshes, in preparation.

[AMN]

R. ABGRALL, K. MER, AND B. NKONGA. A Lax-Wendroff type theorem for residual schemes. In M. Hafez and J.J. Chattot, editors, Proceeding of a conference for P.L. Roe's 60th birthsday. Wiley, to appear.. T.J. BARTH. Some working note on the n scheme. Private communication, 1996. S.F. DAVIS. A rotationaly based upwind difference scheme for the Euler equations. J. Comp. Phys., 56 :65-92, 1983. R. STRUIJS, H. DECONINCK, AND P. L. ROE. Fluctuation splitting schemes for the 2d euler equations. VKILS 1991-01, Computational Fluid Dynamics, 1991.

[Ba] [Da] [SDR]

[Ta]

[DvW]

E. TADMOR. The numerical viscosity of entropy stable schemes for systems of conservation laws. Math. Comp., 49 :91-103, 1987. E. VAN DER WEIDE AND H. DECONINCK. Positive matrix distribution schemes for hyperbolic systems. In Computational Fluid Dynamics '96, pages 747-753. Wiley, 1996.

FIG. 1: (a)-Mesh for the interaction of 2 parallel supersonic flows ; (b)- Density isolines for the supersonic flows. Top-left : N scheme p G [0.7,1.4], top-right : second order MUSCL finite volume scheme p 6 [0.689,1.403] , bottom-left : LDA scheme p 6 [0.615,1.427], bottom-right: present scheme p € [0.698,1.402]

10

Finite volumes for complex applications

FIG. 2: (a)-Mach number isolines for the Naca0012 test case. Top-left : N scheme M G [0.05,1.394] , top-right : second order MUSCL finite volume scheme M <E [0.05,1.422], bottom-left : LDA scheme M 6 [0.05,1.425], bottom-right : present scheme M e [0.04,1.522]; (b)-Reduced entropy £ = •f IN scheme S G [0,0.038],second order MUSCL finite volume scheme S°°e [-0.009,0.039], LDA scheme £ € [-0.009,0.091], present scheme S e [-0.0003,0.032]

A Roe-type Linearization for the Euler Equations for Weakly Ionized Gases

Frederic COQUEL LAN CNRS Tour 55-65 5ieme, U.P.M.C. 75252 Pans Cedex 05 Claude MARMIGNON ONERA, BP 72 92322 Chatillon Cedex

ABSTRACT This paper is devoted to the numerical approximation of the discontinuous solutions of the Euler equations for weakly ionized mixtures of reacting gases. The main difficulty stems from the non conservative formulation of these equations due to a widely used simplifying assumption. We show how to derive a well-posed conservative reformulation of the equations from the analysis of the associated connective-diffusive system. We then propose an exact Roe-type linearization for the equivalent system of conservation laws. Our results can be seen as an extension of the classical Roe average, for nonlinearities that cannot be recast under quadratic form. Key Words: Convective-diffusive systems. Nonlinear hyperbolic systems. Non conservative products. Shock solutions. Roe-type linearization.

1. Introduction

This work treats the numerical approximation of the solutions of a convectivediffusive system, we write for short as

This system governs ionized mixtures of reacting gases in thermal nonequilibrium. Such plasma are studied here in the context of large Mach number flows.

12

Finite volumes for complex applications

The solutions we are interested in, are thus mainly driven by the underlying first order system. The main properties of the extracted first order system

are reported below. This nonlinear system will be seen to be hyperbolic so that its solutions are known to develop, generally speaking, discontinuities in a finite time. But when dealing with discontinuous solutions of (2), a major difficulty arises : there does not exist a flux function, say f, such that A(u) = Vuf (u). In other words, the hyperbolic system (2) is under non conservation form. It is known that the non conservative products involved in A(\i)dxu have no classical sense at the location of a shock since they cannot be given a unique definition within the standard framework of distributions. For this reason, it must be recognized that an additional information is required in order to specify the definition, e.g. the value, of the non conservative product A(u)dxu at shocks. This difficulty has motivated some recent works. We refer in particular to the work by LeFloch [8], DalMaso-LeFloch-Murat [6] where non conservative products are defined on the basis of a fixed family of paths <]? in the phase space : After LeFloch [8] and Sainsaulieu [11], the choice of a particular family of paths $ is dictated by the additional informations brought by the full second order convective-diffusive system (1) (see below for a brief survey). The key feature is that the definition of shock solutions heavily depends on the shape of the diffusive tensor £>(u) which is modeled in agreement with the physics. These definitions provide us with a relevant setting for defining the discontinuous solutions of the non conservative hyperbolic system (2). Once defined, the first order system is well-posed and its numerical approximation could be tackled. However, two difficulties arise in that way. First, a close formula for shock solutions is in general not available. Furthermore, even when explicitely available, we have illustrated [4] that the error in the discrete capture of shock solutions unacceptably grows with the strenght of the shock. We refer to LeFloch-Liu [9] for an error analysis devoted to the scalar case. At this stage, these two difficulties make the numerical approximation of the (strongly) discontinous solutions of (3) to be virtually untractable. To overcome these two difficulties on the same time, we propose to study the existence of a conservative formulation for system (3) that is compatible with the diffusive tensor V. That is to say, we ask for the existence of (at least) one change of variables v = v(u) that brings the non conservative second order

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system (1) (with c«;(ii) = 0) to a fully conservative convective-diffusive system

Let us emphasize that not only the first order system must find a conservation form (PI) but also that the second order operator must stay under conservation form (P2). Actually, these two requirements (P1),(P2) ensure the equivalence of the shock solutions of (3) where <£ and T> are compatible and the shock solutions of We refer to Sainsaulieu-Raviart [12] for a proof. The benefit of such an equivalence is twofold. In a first hand, the shock solutions of (3) are now explicitely given by the Rankine-Hugoniot jump relations associated with (5). In a second hand, Riemann solvers under conservation form can be applied to (5) in order to approximate the equivalent weak solutions of (3). As reported below, a specific family of change of variables turns out to fulfill both (PI) and (P2). For the associated equivalent systems of conservation laws, we then show how to derive an exact Roe-type linearization. 2. Analysis of the extracted first order system

In this section, we focus ourselves on the definition of the extracted first order system (2), the precise shape of the diffusive tensor 1) will be discussed later on. We treat mixture of gases made of electrons and n heavy species, ni, 1 < ni < n of them being ionized. All the species we consider are described with the same mean velocity v. To account for the smallness of the mass ratios Me/M2- « 1, z'G {!,..., n}, the electron gas is endowed with a temperature Te distinct from the temperature T of the heavy species mixture. Moreover, nv, I < nv < n, molecular species have their own vibrational temperature Tvj, j G {1, • • • , nv}. Neglecting at this stage the thermo-chemical relaxation terms, the extracted first order system (2) writes :

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Finite volumes for complex applications

The main properties of (6) stated in the next section will enlighten several relationships with the usual Euler system. Here, Y,, i 6 {1, ...,n} denotes the ith species mass fraction. The conservation laws (6b),(6c) respectively govern the momentum and total energy E of the mixture of the heavy species plus the electron gas. Conservation laws (6e) refer to the vibrational energies pYjevj of the nv molecular species that are assumed to be in thermal nonequilibrium. These (n + nv + 2) conservation laws are supplemented by a balance equation (6d) : the expected conservation law for the electron gas energy pYeEe must be, in fact, balanced by the work of the electric field £. The required closure equations are as follow. The electron gas pressure pe obeys : while the pressure law p for the mixture of heavy species is defined by :

with 7tr G ]l,3j. Here, Cj-(T) refers to the energy of the internal modes of species i at equilibrium with the temperature T and h® denotes its heat of formation. The last closure equation to be specified deals with the electric field £. The expected closure equation for £ should be the Poisson law, written here in a dimensionless form :

In (9), € refers to a parameter proportional to the Debye lenght. This parameter actually yields a rough estimate of the spatial resolution that is required to approximate (9) : namely O ( c — l ) . However in our applications, this parameter turns out to be extremely small (see [14] for instance) To make the numerical approximation tractable, we are lead to let c goes to zero. Doing so, the Poisson law degenerates to the so-called local charge neutrality condition while the closure equation for E is classically given by :

Here, z,- refers to the electric charge of the ith ionized species. The stricking feature of the closure equation (10) stays in that it involves partial derivatives

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and in particular makes £ to be, generally speaking, a measure at the discontinuities of u ! The work of the electric field Neqe£ x v in (6d) is clearly a product involving first order and zero order terms and furthermore, it cannot be put under divergence form. Let us stress that the product Neqe£ x v cannot be understood, as it is commonly handled (see [2] for instance), as a source term, e.g. a function depending solely on zero order terms. On the contrary, such a product is a constitutive part of the first order system.

2.1. Basic properties of the extracted first order system In this section, we state some of the main properties of the first order system (6). The results given below intend to provide a deeper insight into the system under consideration and in particular to shed light on its relationships with the standard 3x3 Euler equations. The phase space Q associated with (6) is the following subset of Rp, p = n + nv + 3 :

We begin with the following result devoted to the smooth solutions of (6). Lemma 1. Let u : R x R+ —>• Q be a Cl solution of (6). Then, the closure equation on the electric field £ reads :

Furthermore, u satisfies the following balance equations in non conservation form :

Dropping the electron gas pressure pe in (13c), the three equations (13b) to (13d) are easily recognized to coincide with the ones governing the smooth solutions of the 3x3 Euler system. Here, the heavy species mixture and the electron gas equally contribute to the velocity balance equation : each with

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Finite volumes for complex applications

their own pressure. The pressure balance equations (13d) and (13c) read the same but the key point stays in their complete decoupling. We indeed have : Proposition 2. i) The first order system (6) is hyperbolic in the phase space Q and A(u) : Q —> Mat(Rp} admits the following three real eigenvalues :

where the eigenvalue v has an order of multiplicity (n + nv + 1). 11) Under standard thermodynamic assumptions, the fields associated with the eigenvalues v ± c are genuinely nonlinear. Hi) The fields associated with the eigenvalue v are linearly degenerate. Their Riemann invariants are given by v and (p + pe). iv) The closure equation (12) for Neqe£ never involves a product of a Dirac mass against an Heaviside function and is thus well-defined. v) The non conservative product Neqe£ x v has no classical sense for the shock solutions of (6). According to the statement v), the shock solutions of (6) stay unknown within the standard framework of distributions. The nonconservative products involved in A(u)dxvi can nevertheless receive a definition in the recent setting proposed in [6]. Indeed, such products can be defined thanks to a given fixed family of paths, denoted $, that are subject to some consistency and smoothness conditions (see [6] for the details). For a shock discontinuity separating the two states u/, and u#, the non conservative product is defined by :

Note that since A is not a jacobian matrix, the definition (16) entirely depends on the choice of <£. After LeFloch [8] and Sainsaulieu [11], the relevant choice of <£ comes from the study of the full second order convective diffusive system (1). Indeed, the shock solutions of (6) can be defined as the limit when the diffusion is neglected of the travelling waves solutions of (1). More precisely, a function u : R x jR+ —>• £1 is a travelling wave solution of (1) connecting the states u/, and UR with speed

Next let be given a small parameter e > 0 and let us consider the function

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u(x f a t } - It can be easily seen that this function is also a travelling wave solution but of the convective-diffusive system (1) with vanishing viscosity :

Furthermore, since d^ue £ L^(R) with ||c/£Ue||£i = ||c^u||£i, we deduce that. {ue} tends almost everywhere as e goes to zero to the step function u := U£-f(u/£ — U L ) H ( 0 ) while for any given e > 0, the followingjump like conditions hold true :

Motivated by the above calculations, LeFloch [8] (see also Sainsaulieu [11]) has defined the discontinuous step function u to be a shock solution of the first order extracted system (2) which is compatible with the diffusive tensor V. It is essential to notice that by contrast with conservative hyperbolic systems, the travelling wave solutions and therefore the step function u depend on the shape of D. Indeed, two non proportional diffusive tensor generally yield distinct shock solutions for the underlying convective system. Such an issue is actually illustrated by the Proposition 8 stated below. The first order extracted system (6) can be closed according to the above framework and its numerical approximation could be tackled. However and as explained in the Introduction, two difficulties arise when dealing with the numerical approximation of the discontinuous solutions of (6). In order to circumvent these two difficulties simultaneously, we have put forward the need for admissible change of varaibles that recast (6) in the full conservation form (4). We indeed have : Proposition 5 (Sainsaulieu-Raviart [12]). Let be given a C1 diffeomorphism W : Q —> £1. Assume that for any given smooth solution u : R x R+ —>• £7 of the non conservative convective diffusive system :

the change of variables v = ^(u) yields a solution of the following SLC :

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Finite volumes for complex applications

Then, the shock solutions of the extracted first order system of (21), e.g. the ones compatible with D, do coincide with the shock solutions of the extracted conservative first order system of (22) :

The benefit of this equivalence is twofold : first the shock wave solutions to (6) are now explicitely given by the Rankine-Hugoniot relations associated with

and in a second hand, upwind methods in conservation form can be applied to (24) in order to approximate the equivalent weak solutions of (6) that are compatible with D. Motivated by these strong benefits, we have laid in a related work [3] the foundations for a systematic characterization of all the CldifFeomorphisms ^ that bring (22) from (21). The starting point stems from the following observation : the equivalence stated in (23) requires in fact the fulfilment of two distinct conditions. Namely, not only the first order system must meet a conservation form (P1) but also that the second order operator must stay under conservation form (P2). In what follows, we study for existence the admissible changes of variables ^ that satisfy both requirement (P1) and (P2). 2.2.1. Fulfilling (P1). In order to exhibit the C 1 -diffeomorphismsthat satisfy the requirement (P1), we propose to characterize in a first step all the additional scalar conservation laws satisfied by the smooth solutions of (6). Each additional conservation law indeed yields an obvious change of variables by substitution with the single equation in non conservation form (6d). We specifically have : Theorem 6 Let u : R x /?+ —>• Q be a Cl solution of (6). Then u satisfies all the additional (non trivial) scalar conservation laws :

where g : £1 —)• R denotes an arbitrary (say of class C1) function of the common Riemann invariants of the two genuinely nonlinear fields : e.g.

Assume that

d Q

? Te ^ 0, then the Cl diffeomorphismm^g

M/ p (u) = ({pYj}, pv, pE, pg(\i), pYjeVtj)

: £2 —)• f2, u —>

yields the smooth solutions of (6)

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to be equivalent smooth solutions of the following hyperbolic system in conservation form :

The above family of C^diffeomorphisms that satisfy (PI) for the smooth solutions of (6), might look rather large. However, the next statement shows that all these possible candidates for the fulfilment of both (PI) and (P2) are indeed closely related. Corollary 7. i) All the additional conservation laws specified in (26) can be recovered from the following finite set of transport equations (since dtg(u) + vdxg(u) = 0;

n) The functions g : Q —>• jR are Riemann invariants for the two genuinely nonlinear fields and thus stay constant across the associated rarefaction waves, iii) Assume that one of the change of variables ^g specified in (28) meets the requirement (P2). Then, necessarily the related function g(u) stays continuous for (so constant across) the shock solutions of (6) that are compatible with D. The consequences of properties i) to iii) are briefly discussed in the next section since they will serve as a natural guideline for investigating for validity (P2) with all the possible candidates Wg. If one of them turns out to satisfy (P2), the resulting conservative system (28) and the extracted first order system (6) will not only describe the same smooth solutions but will also admit the same discontinuous solutions : namely, (24) will hold true. This issue is addressed below. 2.2.3. Fulfilling (P2). The existence of a C 1 -diffeomorphism that obeys (P2) obviously heavily depends on the shape of D. In view of iii), Corollary 7, its shape must be compatible with the conservation across the shock-solutions characterized by

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D of a least one function g(u) in the form (26). According to the Definition 4, the characterization i), Corollary 7, leads us to derive the travelling wave equations that govern (Yi, in(p//?7), ln(pe/p~is), Yjevj) in the presence of V and to study the nonlinear combinations in the form (26) that yield a fully conservative equation. The analysis [4] shows that none of the \&g given in (27) can achieve such a requirement for a general diffusive tensor D. However, this negative result can be bypassed provided that a physical assumption is made. This is the matter of the next statement. Proposition 8. i) Let the diffusive tensor D(u) be defined by

where v, K respectively denote the viscosity and the thermal conductivity of the heavy species mixture, Ke the thermal conductivity of the electron gas and KVJ the one of the jth molecular species in vibrational nonequilibrium. Then, the associated travelling wave u satisfies for all t; £ R, with M = p(£)( v (£) — cr) =

Furthermore, for arbitrary (*/, K, Ke, KV,J], none of the Cl-diffeormophisms specified in (27) does satisfy the property (P2). ii) Assume that Ke — 0, e.g. that the diffusive tensor £>(u) is now given by :

Then, all the Cl diffeomorphisms tfg based on 7e for the electron gas stays constant accross the shock waves that are compatible with (30). We underline that such a consequence is indeed compatible with the thermodynamic second principle : the heavy species specific entropy p/p^ indeed strictly increases accross shock waves (see also n) below). The physical interpretation of these admissible Cldiffeomorphisms can be found in Zeldovich-Raizer [14] (vol. 2). Proposition 8 provides us with a whole family of \&g that fulfill both (PI) and (P2).

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3. A Roe-type linearization

In this section, we propose a Roe method for the conservative hyperbolic system (33) in two space-variables. Since these equations stay invariant under rotations, the scheme makes use of the classical dimensional splitting and is thus derived for the associated quasi-lD system, we write with clear notations :

In what follows, we show how to derive an exact Roe-type linearization on the basis of an original Lemma for averagings. This Lemma will turn out to be central in the handling of the highly nonlinear pressure law pe. We just recall that a Roe-type linearization [10], we denote -B(v/,,V.R), has to satisfy the following three requirements:

In (37), the jacobian matrix V v ^ r (v): Q —> Mat(Rp} of the exact flux can be expressed in terms of the following set of arguments :

The precise shape of V v J r (v) is given in a companion paper [4]. Motivated by recent works [1], [13] devoted to mixtures of neutral gases, e.g. without ionization effects ; we seek for a linearization considering the following averaged form for V v ^ 7 (v) :

We have to define the set of averages involved in (39) so that the three conditions (37a)-(37c) hold true. Besides the hyperbolicity condition (37a), (37b) requires that such averages have to be derived so that the jump AJ r (v), which is of course nonlinear in v, can be reconstructed from the linearized form (39) of V v ^ r (v) times the jump Av. It is worthy to recall that for the 3x3 Euler system, the classical Roe average (see below) basically stems from the existence of a set of variables, the so-called parameter vector [10], for which all the

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nonlinearities involved in the system can be recast under quadratic form. Furthermore, within this framework, this Roe average turns out to be the unique average that yields (37a)-(37c). In our setting, some of the scalar conservation laws involved in (35)-(36) actually exhibit the same type of quadratic nonlinearities. We are thus lead to endow the related variables with the Roe averaging procedure : namely, we choose

But the pressure laws we have to deal with, bring a different type of nonlinearities. Averaging their partial derivatives indeed requires a specific treatment such that, besides (37a), the following consistency conditions hold true :

In fact, within the framework of neutral gases, Vvp has been already given suitable averaged forms. We refer in particular to the works by Abgrall [1] and Shuen-Liou-VanLeer [13]. When ionized species occur, we have shown [4] that the first identity in (41) can be derived from the framework of neutral gases provided that the second identity is valid. Our Roe-type linearization tacitly makes use of one of these relevant averages for V v p. From now on, we focus our attention on the derivation of an averaged form for V v p e which is consistent with the three requirements (37a)-(37b). In that aim, we state the following first result : Proposition 11. i) The averaged gradient V v p e can be recontructed from a set of three averaged partial derivatives :

where

if the

species i ionized and 0 otherwise, Next, assume that is associated with

and Vvp derived from [1] or [13]. Then, assumption (H) yields : ii) the Roe consistency condition (37b) is met iff for any given (v/,, v/?) £ ^2

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in) Whatever are these three averages, B(VL,VH] admits the set of eigenvalues (v — V c 2 , v, v + V c 2 ). Moreover, the real number c2 stays positive provided that for any given (v^, v#) G Q2, the following, where (7,7 e ) G ]1,3] 2 , fto/c/s £rue :

In order to enforce the validity of the two conditions (44) and (45), we now consider the following easy but key lemma for averagings (see also Abgrall [1], Godlewski-Raviart [15]). Averaging Lemma. Let be given (XL, XR), (yi, UR) two pairs of real numbers. Let (77,, TR) be any given pair of real numbers such that TL + TR ^ 0. Let us define the following unsymmetric r-averaging operators :

Then, the following identity holds true :

Let us apply for instance the above averaging Lemma to the pairs of interest (/>, X] with the £-averaging we define by gL=^/PL, QR — ^fpR• We easily get from (46) and (47) the well-known Roe identity, we write with classical notations :

Turning back to the general case, we emphasize that the identity (47) is indeed valid for any given pair (TL, TR). Taking advantage of such a degree of freedom, we introduce below a (wide !) family of unsymmetric averagings that makes always valid the required consistency condition (44). Equipped with these families, we next turn studying how to enforce the hyperbolicity condition (45). To that purpose, the specification of the underlying averaging operator is clearly the central issue. The statements, given below, summarize our main results. They are intended to shed some light on the application of the averaging Lemma we have introduced in order to enforce the validity of both (44) and (45).

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3.1. Enforcing the Roe consistency condition (44)In view of the electron gas pressure law (34), the required discrete identity (44) writes :

The above consistency condition involves increments in the conservative variables pYe, p and se while the underlying nonlinearities merely occur in terms of the primitive variables Ye, p and se/p. To make the needed linearization tractable, we specify in the next statement a convenient family of averages for (42) that shifts (44) from the conservative to the primitive variables. Proposition 12. Let us consider the following averaged forms for the partial derivatives (42) of the electron gas pressure law :

Then, for any given pair of states (VL, v#) ; the consistency condition (44) zs equivalent to the following identity, where p is defined in (48) :

In order to satisfy (51) and thus (44), it remains to define the unspecified mean values in (50). This is the matter of the next statement. Proposition 13. With the notations of the Averaging Lemma, let us define the averaged forms in (50), using respectively an arbitrary a-averaging operator :

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and an arbitrary (^-averaging operator :

Then, the identity (51) and thus the Roe consistency conditions (37b)~(37c) are always valid. In what follows, we shall assume for convenience that both a and (" stay non negative. Equipped with this wide family of relevant averagings, we now turn studying for validity the hyperbolicity condition (45). 3.2. Enforcing the hyperbolicity condition (45) The main result of this second step is as follows : Proposition 14. Let us respectively define the a and (, averaging operators in (52), (53) by

Then, (54), (55) provide us with the unique pair of averaging operators such that the hyperbolicity condition (45) holds true for any given (VL, v#) £ Q2. We have the following final statement. Theorem 15. Let us consider the following averaged form of V v ^ fr (v) :

with [YI, v, w, H, se/p, YjeVtj} given in (40), with Vvp derived from [1] or [13] and with V^pe constructed from (40), (50), (52)-(53). Then, for any given pair of states (VL, v#) € Q2, B(VL, VR) admits three real eigenvalues with a complete set of right eigenvectors. Moreover, this matrix satisfies the Roe consistency conditions (37b) and (37c). Therefore, B(VL, v#)

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is an exact Roe-type linearization for the system (35)-(36). We emphasize that the averaging operator built on (55) provides one with an extension of the classical Roe-average (48). Indeed, let us compare two of the nonlinearities that arise in the pressure laws : namely p7e x {h(^}} with P x (?(~) }• Now, it is sufficient to apply to the latter nonlinear expression our formula (55) with the associated exponent a = I in place of je to get v/p. [I]

R. ABGRALL, La Recherche Aerospatiale, vol 6, 31-43, 1988.

[2]

G. CANDLER AND R.W. MACCORMACK, AIAA paper 88-0511, 1988.

[4]

F. COQUEL AND C. M A R M I G N O N , Work in preparation.

[5]

S. CORDIER ET a/., Asymptotic Analysis, vol 10, 1995.

[6]

G. DALMASO, P.G. LEFLOCH AND F. MURAT, J. Math. Pure Appl., vol 74, 483-548, 1995.

[7]

B. LARROUTUROU, Computational methods in applied sciences, ECCOMAS, Eds Ch. Hirsch, Elsevier, 1992.

[8]

P.G. LEFLOCH, IMA Preprint series No 593, University of Minnesota, 1989.

[9]

P.G. LEFLOCH AND J. LIU, Math, of Comp., 1994.

[10]

P.L. ROE, J. Comp. Phys., 357-372, 1981.

[II]

L. SAINSEAULIEU, SIAM J. Appl. Anal., 1995.

[12]

P.A. RAVIART AND L. SAINSEAULIEU, Mathematical Methods and Models in Applied sciences, 1995.

[13]

J.S. SHUEN ET a/., NASA TM 100856, 1988.

[14]

YA. ZEL'DOVICH AND Yu. P. RAIZER, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomen, vol 1 and 2, Academic Press, 1966.

[15]

E. GODLEWSKI AND P. A. RAVIART, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer (Eds), Applied Mathematical Sciences, vol. 118, 1996.

Multidimensional upwind residual distribution schemes and applications

H. Deconinck and G. Degrez von Karman Institute for Fluid Dynamics Sint-Genesius-Rode, Belgium

ABSTRACT A review is made of upwind residual distribution schemes (RDS) for hyperbolic systems for application to compressible and incompressible flows. First, the design principles of RDS are explained for the simplest case of linear scalar advection. Extension to linear hyperbolic systems is described next. Then, their application to compressible and incompressible flows is discussed, and illustrative examples of applications are presented. Key Words: finite volume method, finite element method, compressible flows, incompressible flows

1.

Introduction

Multidimensional upwind residual distribution schemes, which were first introduced by P. L. Roe [ROE 87], have been developed on ideas borrowed from both the finite volume and finite element methods to become nowadays an attractive alternative to either one [PAI97]. The initial motivation for their development was a discontent about some drawbacks of the state-of-the-art finite volume solvers based on 1-D approximate Riemann solvers, namely • D by D first order upwinding is very diffusive, • ID Riemann solvers do not capture the real multidimensional flow physics, • higher order schemes use wide stencils. The starting point for the development of these schemes was a reinterpretation of ID finite volume schemes based on the concept of fluctuation [ROE 82]. Considering the continuous piecewise linear data representation classically used in finite element methods rather than the discontinuous cell-wise reconstructions used in finite

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volume methods, the flux difference between two nodes (F/+1 — F() is reinterpreted as the flux balance (else fluctuation or residual) over the interval (element) [i,i+ 1] (= fedF/dx dx), which is to be distributed to the vertices of the element. This interpretation can be carried over to several dimensions, provided the fluctuation is now the flux balance over a simplex (triangle/tetrahedron) to which is associated a continuous piecewise linear data representation. Because the elementary discretisation unit is an element rather than an edge or face for the finite volume method, the multidimensionality of the problem can genuinely be taken in to account: in 2D, the fluctuation can be distributed to the three vertices of the triangle rather than to the two neighbouring cells of an edge. Based on this idea, schemes were constructed, which combine a number of attractive features: • a much lower cross-diffusion than their finite volume counterparts, due to the genuinely multidimensional upwinding they incorporate, • a positivity property which ensures the satisfaction of a discrete maximum principle and consequently the absence of spurious oscillations, • (almost) second order accuracy on compact stencils. In addition, the compact discretisation stencil allows for the development of efficient implicit iterative solution strategies [ISS 96] and for an easy parallelisation [ISS 98, vdW99]. The paper starts with the presentation of upwind residual distribution schemes for linear hyperbolic scalar equations and systems, successively. This is followed by a discussion on their application to non-linear problems, specifically to the compressible and incompressible flow equations. Finally, the paper concludes with a few illustrative computational examples.

2.

Linear equations

2.1. 2.1.1.

Scalar advection Design principles

The residual distribution schemes (RDS) have been designed for an optimal discretization of the steady state convection equation

on PI finite element meshes, i.e. triangular (resp. tetrahedral) meshes with a piecewise linear solution representation. Evaluating the residual or fluctuation over an element

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Figure 1: Triangular element with scaled inward normals T, defined as the integral over this element of the differential operator, i.e.

since both /L and VM are constant over the element, one obtains §T — £,-&,-M,-, where ki = ^ A • nt is called the inflow parameter, ni being the scaled inward normal of the edge opposed to node i (see Fig. 1) and d the dimension of the problem. The method consists in distributing fractions of r to the vertices of the element. The resulting discrete equations therefore express that the nodal residuals Rf, sum of all contributions from neighbouring elements, vanish, i.e.

in which j8-r are the distribution coefficients. On each element T, these distribution coefficients must sum up to one for consistency and conservativity. The different schemes, corresponding to different ways of computing the distribution coefficients, have been designed to satisfy several properties making them optimal: UPWIND CHARACTER (W): No fraction of the element residual is sent to upstream nodes or /3^ = 0 when k{ < 0. POSITIVITY (^): The scheme does not create local extrema or, if we write the conT tribution to the element residual as 6? = Qj6 = y^J/ c -•• J« ,J, we impose c,,lj <— ' * 0 V; ^ i.

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Scheme N LDA SUPG PSI

Table 1 : Residual distribution schemes type <% K linear V W'Xy+-*7)-%*7K--K;) + linear %/ 1 , kj_ 1 h linear T T

(XA )A

(%>

t^S

V

-2nX||

max(0,)3/ v )/I I .max(0,j3/ v )

non-linear

V V

JS?^ V V V

LINEARITY PRESERVATION (jSf £*): The distribution coefficients jSf remain bounded as 0r —> 0 which is equivalent to zero cross-diffusion (second order accuracy) on a regular mesh. CONTINUITY CT^): The distribution coefficients must be continuous functions of both the advection and the solution-gradient directions. Several schemes satisfying some or all of these design criteria have been developed both in 2D and 3D. A complete summary is given in [PAI 97]. The schemes used in the present study are summarized in Table 1 (formulas valid both for 2D and 3D), where k+ = max(0,& ( ), &r = min(0,/c / ). Note that as a result of a generalization of Godunov's theorem, only non-linear schemes can satisfy both the & and %*& properties. 2.7.2.

Finite element interpretation

Residual distribution schemes can be viewed as generalised Petrov-Galerkin finite element schemes. Indeed, denoting wj the weighting function associated to node i of element T, the Petrov-Galerkin discretisation of the convection equation (1) can be written

Now, from the definition of r and from the linear approximation of u over T, assuming a constant advection vector A,, A- • VM is constant over T and equal to 0 r /£1 T , where Q.T is the surface of the triangle T (volume of the tetrahedron T in 3D). Hence, the previous equation becomes

For the Petrov-Galerkin discretisation to be identical to the residual distribution discretisation (3), the weighting functions co[ should satisfy the following relation :

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This equation is not sufficient to uniquely associate a weighting function &)? to a residual distribution coefficient /3(^ unless the functional form of the weighting function is prescribed and depends on a single parameter. Prescribing the weighting function to be the sum of the finite element shape function (jp- and a piecewise constant contribution similar to SUPG weighting functions [DEM 97], i. e. (of = cpt + aj yT, where aj is the upwind bias coefficient and JT is the piecewise constant function equal to 1 on element T and 0 elsewhere, the equivalence condition (6) yields

2.2.

Hyperbolic systems Considering now the non-commuting linear hyperbolic system

the formal extension of the residual distribution discretization (3) is

where the fluctuation Or is now a vector whose expression is Or — £.K(.U(. with K( being now an inflow matrix expressed as K- = ^ A^n (f, and where fl? is a distribution matrix. The design criteria for distribution matrices are the same as for distribution coefficients for the scalar advection equation, plus the additional requirement of invariance under a similarity transformation, i. e. UPWIND CHARACTER: j3(r = 0 when K,. < 0 (where K. < 0 means that all eigenvalues of K,, which are known to be real due to the hyperbolic nature of the system, are negative). r r r J condition becomes C,. < POSITIVITY: With Of' = /3 *«; O = Y. *^j C;ij; U ;j, the positivity 'j — 0 V; / i.

LINEARITY PRESERVATION: The distribution matrices ff remain bounded as*r — 0. CONTINUITY*

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INVARIANCE: Under the similarity transformation <9U = TdQ, the hyperbolic system (8) transforms into A^|^ = 0 where Ag£ = T~1A.£T. The discretization of the transformed equation being

we require that It results that if the system is diagonalisable, i. e. if the matrices A^ commute, then taking T as the diagonalising transformation, the transformed distribution matrix )3l- is the diagonal matrix of scalar distribution coefficients for each decoupled equation. Many of the schemes listed in Table 1 can be formally extended to systems. For example, the system-N scheme is defined by

whereas the system-LDA scheme is defined by

The non-linear PSI scheme on the other hand proves to be more difficult to generalize. Considering the scalar PSI scheme as a limited N-scheme, its formal generalization to systems is

However, the distribution matrix of the N-scheme f$7'N is not explicitly defined by Eqn. 11. For diagonalizable systems, the condition of invariance under similarity transformations suffices to define ($T'N and hence ^'PSI uniquely. For general noncommuting systems, the invariance condition is no longer sufficient. The system PSI scheme used in the numerical applications is based on one particular definition of [3T'N which satisfies this condition. Further details are given in [PAI 97]. The construction of non-linear positive and linearity preserving schemes for systems is still an ongoing research topic. New developments, based on the reinterpretation of the scalar PSI scheme as a blended N/LDA scheme are discussed in R. Abgral1's invited paper in this conference [ABG 99].

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Compressible flows

3.1.

Conservative linearisation

The residual distribution schemes presented in the previous section have been defined for linear problems. To apply them to non-linear problems such as the Euler equations, an essential ingredient is a conservative linearization, which consists in finding on each triangle T an average state U such that

where

so that, locally, the non-linear system of conservation laws d^^/dx^ = 0 is approximated by the linear system A^dU/dj^ = 0. Let's show that, with such a linearization, conservation is indeed satisfied. Summing up the nodal residuals over the domain £1, we have

and the contributions of internal edges cancel out (telescoping property). For the Euler equations, a conservative linearization is easily obtained as a multidimensional extension of Roe's linearization [DEC 93]. Indeed, assuming a linear variation of Roe's parameter vector Z [ROE 81] and since both the vector of conserved variables U and the fluxes F^ are homogeneous functions of degree 2 in the components of Z,

where Z = ^rf Svertices^i ^s ^e Pr°Per average state to ensure conservation. 3.2.

Transformations and preconditioning

The matrix distribution schemes presented above are invariant under a similarity transformation. It is nevertheless useful to apply a similarity transformation to the lin-

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earized hyperbolic system in order to achieve maximum decoupling for the following reasons. On one hand, this reduces the computational cost thanks to the considerable simplification of the distribution expressions and on the other hand, this allows to select different distribution schemes on the various decoupled systems/scalar equations. Specifically, with the similarity transformation dU = TdQ, the discrete equations (9) are rewritten

where O^ = Z,-Kg(.Q.. Partial decoupling can be achieved if the flux jacobians A^ have common eigenvectors. For the Euler equations, the flux jacobians A^ have one common eigenvector so that it is possible to decouple one scalar equation from the original system, leaving a coupled 3x3 system and one decoupled scalar equation in 2D. The transformation to symmetrizing variables defined as dQ = (-^,du,dp a2dp}1 accomplishes this task. The decoupled scalar equation is nothing else than the entropy advection equation, which is well-known to derive from the Euler equations. As shown in [PAI 97], additional decoupling may be achieved by preconditioning, namely the system of equations is rewritten as1

and the residual distribution method is applied to the preconditioned system between brackets. The optimal preconditioning was found to be the van Leer-Lee-Roe preconditioning [vLE 91], which allows to decouple one additional equation, namely the total enthalpy advection equation. 3.3.

Viscous terms

Viscous terms are discretised using the finite element interpretation of the schemes presented in section 2.1.2.. Specifically, the space discretisation of the divergence of the stress tensor V • t is

Now, since

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4. Incompressible flows For application to incompressible flows, two avenues have been explored. The first approach, followed by Waterson & Deconinck [WAT 97], is based on the fact that the incompressible Navier-Stokes equations are a symmetric advective-diffusive system [DEM 97], which can therefore be discretised using the system distribution schemes presented in section 2.2. using a collocated PI finite element representation for all variables. The collocated arrangement is made possible by the inherent pressure stabilisation effect of the system discretisation. The second approach, followed by Bogaerts et al. [BOG 98], is based on a stable finite element representation of the velocity and pressure fields (PlisoP2/Pl element) which therefore does not require any pressure stabilisation. It results that scalar residual distribution schemes can be used to discretise the momentum equations, i.e. using the finite element interpretation, the discretised equations are

where (oik is the residual distribution weighting function associated to the velocity node / and to the k\h component of the momentum equation, and i// is the shape function associated to the pressure node j. Note that, because of the upwind component of the weighting function co- k, upwinding is introduced in the discretisation of the pressure gradient. This turns out to be essential for the accuracy of the method [BOG 98]. 5.

Applications

Capabilities of upwind residual distribution schemes are now illustrated by a couple of computational examples. 5. /.

Inviscid transonic flow over the M6 wing

The ONERA M6 wing is a well documented testcase for three-dimensional flows from subsonic to transonic speeds [AGA 94]. The selected transonic case is Moo = 0.84, a = 3.06°. The grid consists of 316275 nodes and 1940182 tetrahedra. The far-field boundary is half a sphere with a radius of 12.5 root-chord lengths. The computation of this testcase with the present multidimensional upwind method employed the van Leer-Lee-Roe preconditioning, allowing a hybrid discretization. The decoupled

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advection equations for entropy and total enthalpy are discretized with the scalar PSIscheme, while the acoustic subsystem is discretized with a blended system LDA/Nscheme. To give an indication of how the multidimensional upwind method performs compared to standard finite volume methods, the same testcase has also been computed with the Jameson-Schrnidt-Turkel (1ST) artificial dissipation scheme, a matrix dissipation (MATD) scheme and the Roe upwind scheme with Venkatakrishnan's reconstruction. Plots for the pressure coefficient Cp and the relative total pressure loss, defined as 1 — pt/'ptoo, in the 80% span cross-section are shown in Fig. 2. From these figures it is clear that the multidimensional upwind method is much less dissipative in the leading edge region and that it has a better shock capturing. 5.2.

Inviscid transonic flow over a complete aircraft

To demonstrate the applicabilty of the present multidimensional upwind method to a complete aircraft configuration, the transonic flow over a generic model of the Falcon 2000 executive jet has been computed. The grid for a half model consists of 45387 nodes and 255944 tetrahedra. The selected testcase corresponding to the cruise condition is M^ — 0.84, a = 3.06°. The solutions computed with monotone first and second-order multidimensional upwind schemes are compared. For the first order computation the system N-scheme is applied directly to the full Euler equations. The second order computation employs the van Leer-Lee-Roe preconditioning, where the advection equations for entropy and total enthalpy are discretized with the scalar PSIscheme and the acoustic subsystem is discretized with the blended system LDA/Nscheme. Figures 3-4 compare the Mach number and entropy isolines for these two solutions. The better shock capturing and lower spurious entropy generation in the leading edge region of the wing and the engine pylon are clearly observed.

5.3.

Incompressible turbulent flow over a backward-facing step

We consider now the incompressible turbulent flow over a backward-facing step, experimentally studied by Kim [KIM 78]. Calling H the step height, the inlet channel width is 2H, so that the outlet channel width is 3H. The flow Reynolds number is 1.41 105 based on the inlet centreline mean velocity t/0 and the outlet channel width. The standard k — e turbulence model [JON 72] is used with wall function boundary conditions. The grid is a triangulated stretched Cartesian grid extending from 3H upstream of the step to 21H downstream. Along all solid walls, the computational domain boundary is set at a distance h — 0.025// from the wall. The grid contains 6247 PI nodes distributed along 107 vertical grid lines, where each vertical grid line contains 41 PI nodes in the inlet channel and 61 PI nodes in the outlet channel. The inlet boundary conditions for u, v and k are taken from the experiment and £

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Figure 2: M6 wing: Cp and total pressure loss distributions at 80% span, upwind residual distribution schemes versus finite volume schemes

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Figure 3: Falcon: Mach number contours

Figure 4: Falcon: Mach number contours is evaluated using a mixing length model. At the outlet, p is set to 0 and homogeneous Neumann (fully developed flow) boundary conditions are specified for all other variables. Along the wall function boundaries, a homogeneous Neumann condition is used for k and a Dirichlet condition is used for e (= c3/ 4 fc 3 / 2 / k h ). The wall shear stress is derived from the law of the wall. Fig. 5 show the flowfield pattern calculated using the LDA scheme. The reattachment length is 6.3H, a value closer to the experimental value of Lr/H — 7.0 than

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Figure 5: Turbulent flow over a backward-facing step: mean flow streamlines values obtained by other numerical computations using the same turbulence model, which fall in the range (5.7 < Lr/H < 6.0 [DEM 97]). The present method is also seen to capture a small secondary recirculation region at the foot of the step. Acknowledgements This paper summarises over 10 years of research at VKI, which was mainly carried out by a series of Ph.D. candidates: R. Struijs, H. Paillere, E. van der Weide, J.-C. Carette, E. Issman, N. Waterson, S. Bogaerts, K. Sermeus. The ONERA M6 wing and the generic Falcon computations were carried out within the collaborative IMT project IDeMAS funded by the European Commission. The reference computations of the ONERA M6 test case were performed by Daimler-Chrysler Aerospace. The grid for the generic Falcon configuration was provided by Dassault Aviation. References [ABG 99] R. Abgrall. Construction of genuinely multidimensional upwind schemes. 2nd International Symposium on Finite Volumes for Complex Applications, July 1999. [AGA 94] A Selection of Experimental Test Cases for the Validation of CFD codes. AGARD Advisory Report No 303, 1994. [BOG 98] S. Bogaerts, G. Degrez, and E. Razafmdrakoto. Upwind residual distribution schemes for incompressible flows. Fourth European Computational Fluid Dynamics Conference, Athens, Sep. 1998. [DEM 97] T. De Mulder. Stabilized finite element methods for turbulent incompressible singlephase and dispersed two-phase flows. PhD thesis, K. U. Leuven, Leuven, Belgium, 1997. [DEC 93] H. Deconinck, P. L. Roe, and R. Struijs. A multidimensional generalization of Roe's

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Finite volumes for complex applications flux difference splitter for the Euler equations. Computers and Fluids, 22:215-222, 1993.

[ISS 98]

E. Issman and G. Degrez. Non-overlapping preconditioners for a parallel implicit Navier-Stokes solver. Future Generation Computer Systems, 13:303-313, 1997/1998.

[ISS 96]

E. Issman, G. Degrez, and H. Deconinck. Implicit upwind residual-distribution Euler and Navier-Stokes solver on unstructured meshes. AIAA Journal, 34(10):20212029,1996.

[JON 72] W. Jones and B. Launder. The Prediction of Laminarization with a Two-Equation Model of Turbulence. Int. Journal of Heat and Mass Transfer, 15:301-304,1972. [KIM 78] J. S. Kim. Investigation of separation and reattachment of a turbulent shear layer: Flow over a backward-facing step. PhD thesis, Stanford University, Stanford, Ca, 1978. [PAI 97]

H. Paillere, H. Deconinck, and E. van der Weide. Upwind residual distribution methods for compressible flow: an alternative to finite volume and finite element methods. VKI LS 1997-02,1997.

[ROE 81] P. L. Roe. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of computational physics, 43:357-352,1981. [ROE 82] P. L. Roe. Fluctuations and signals, a framerwork for numerical evolution problems. In K. W. Morton & M. J. Baines, editor, Numerical Methods for Fluid Dynamics, pages 219-257. Academic Press, 1982. [ROE 87] P. L. Roe. Linear advection schemes on triangular meshes. CoA Report 8720, Cranfield Inst. of Tech., 1987. [vdW 99] E. van der Weide, H. Deconinck, E. Issman, and G. Degrez. A parallel, implicit, multidimensional upwind residual distribution method for the Navier-Stokes equations on unstructured grids. Computational Mechanics, 23(2): 199-208, 1999. [vLE 91]

B. van Leer, W. T. Lee, and P. L. Roe. Characteristic time stepping or local preconditioning of the Euler equations. AIAA Paper 91-1552-CP.

[WAT 97] N. P. Waterson and H. Deconinck. A fully-implicit multidimensional upwind approach for the incompressible Navier-Stokes equations. In C. Taylor, editor, Numerical methods in laminar and turbulent flows, volume X. Pineridge Press, 1997.

Overcoming mass losses in Level Set-based interface tracking schemes

Th. Schneider and R. Klein Konrad-Zuse-Zentrum fur Informationstechnik Berlin, Germany FB Mathematik & Informatik, Freie Universitdt Berlin, Germany

ABSTRACT An extended level set method is presented for tracking material interfaces in incompressible two-phase flow that ensures conservation of mass. Inconsistencies, between mass transport and interface (level set) motion due to truncation errors are reconciled by two correction steps. Step 1 involves a redistribution of mass from cells that are not intersected by the tracked front but carry unphysical intermediate densities. Step 2 controls deviations between level set and density based partial volume fractions by introducing a small correction velocity in the level set transport equation. Key Words: level set, VOF, interface tracking, incompressible two-phase flow

1. Introduction Among the approaches to compute incompressible flows with material interfaces such as level set methods, volume of fluid methods (VOF) and interface tracking schemes, level set methods currently attract considerable attention. Since level set methods only require solution of a scalar hyperbolic transport equation, they are simple from an algorithmic point of view, their implementation is straight forward and the computational effort to solve the scalar transport equation is negligible in comparison to the effort an incompressible method requires. Furthermore, level set methods naturally support interface distortion as well as topological changes of the interface. The major drawback of level set methods is that they do not ensure conservation of mass by construction. Volume of fluid methods consist of two parts : reconstruction of the interface and advection of the volume fraction. The first part implies a considerable algorithmic effort since the only available information concerning the position of

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the front is hidden in the distribution of the volume fractions. By construction volume of fluid methods conserve mass; in fact for piecewise constant densities, the volume fraction transport equation is equivalent to mass conservation. This is not true in more general cases, though.

2. Level set methods

In the framework of tracking material interfaces in incompressible flows the underlying idea of the level set approach is to identify the phases by the sign of a scalar field G. In other words : areas where G > 0 are occupied by phase I, whereas areas where G < 0 are occupied by phase II. Therefore, the location and the topology of the interface are described by the zero level set of G. Since the interface is advected along particle paths the evolution equation for G is:

Sussman et al. [Sus 94] were the first to use a level set formulation to represent interfaces in incompressible flow. They performed computations of incompressible two-phase flows with density ratios up to 1000:1; they observed considerable mass losses. In order to reduce mass losses they reinitialized G after each time step. Their reinitialization ensures that G remains a distance function, i.e. ||VG||2 = 1. This is achieved by performing a pseudo time integration of

until a steady state is reached. The crucial information that is needed from the distribution of G is the zero level set that represents the interface. In areas where G > 0 or G < 0 the scalar field G is, up to its sign, physically meaningless. This implies that any change may be applied to G in those areas as long as the sign of G remains unchanged. Adalsteinsson et al. [Ada 99] proposed a numerical method to solve the transport equation (1) of G by constructing appropriate extension velocities such that G remains a distance function. Chang et al. [Cha 96] used another reinitialization that ensures conservation of area and by that conservation of mass. They reinitialize G by integrating

until a steady state is reached. Herein e is a small positive constant, K, is the local curvature of the front. Even though this ansatz overcomes the key

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problem of loss of volume when the advecting velocity field is divergence free, it is insufficient for our purposes: The ultimate goal of the present efforts is to construct a flow solver for general zero Mach number variable density flows with phase interfaces. Thus we must combine the level set description with a discretization of mass conservation. This is not achieved in [Cha 96]. 2. Volume Of Fluid Methods

A detailed description of numerous variants of VOF methods can be found in Rider et al. [Rid 95], [Rid 95/2] and [Rid 98]. All VOF methods have in common that they consist of two parts: 1. Reconstruction of the interface : Since no direct information on the topology (normal vector n, curvature K) of the interface is available at each time step the interface needs to be reconstructed based upon the distribution of the volume fractions / in order to allow the advance in time. 2. Advection of the volume fraction /: The volume fraction / is advected according to the following equation:

The flux F is constructed using the information of the interface motion that is obtained in the first part; V is an arbitrary control volume, e.g. grid cell. The numerical method that is used to solve the advection must ensure that / remains bounded : 0 < / < 1. Therefore, if this latter condition is violated mass will be redistributed to mixed cells in the neighbourhood that can absorb it.

3. A coupled Level-Set/Volume-Of-Fluid Algorithm

Bourlioux [Bou 95] presented a coupled level set/volume of fluid method. Her basic idea was to solve an advection equation for the volume fraction / in a VOF manner and the scalar G at the same time. The geometric information of the interface needed to advect / is obtained from a scalar field G; therefore, a reconstruction of the interface from the VOF step function data is not required. During the reinitialization of G she applied an additional correction to

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G in order to minimize the difference of the volume fraction / and the volume fraction fa that results from the integration of the distribution of G within a cell according to

H(G) denotes the Heavyside function.

4. A coupled Level Set / mass conservation Scheme Our proposed method is different from Bourlioux's work in many respects even though the basic idea remains the same. The advection equation of the volume fraction / can be written as follows (the volume fraction f is associated with the phase to which the positive sign of G is assigned) :

H(G) denotes the Heavyside function. If the control volume V is a grid cell with HI cell interfaces a first order discretization of the above advection equation of the volume fraction / is :

n+2

Where /^

is a second order approximation for :

Equation (7) does not automatically ensure that fn+l remains bounded 0 < fn+l < I which results from truncation errors. Therefore, a correction A/ n+1 needs to be added to /n+1. In order to conserve mass A/n+1 is introduced as correction of the fluxes in equation (7) according to :

The equation above is a discrete Poisson equation for the scalar that establishes an implicit coupling between all mixed cells along the interface. Using

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this correction (/n+1 - fn+1 + A/ n+1 ) equation (7) changes to:

A second correction is applied to the scalar field G in order to control the difference of the partial volume fraction fn+l and f£+l- The idea is to locally move the front normal to itself in order to minimize this error. To determine a local correction velocity s* a piecewise linear reprensentation of the front is assumed. The local correction velocity is set to s* — (h — ho)/At (see sketch below): This local correction velocity s* is used to perform one time step of the

Figure 1: Piecewise linear representation of the front within a mixed cell following equation:

5. Results 5.1 Advection of a circle

To test the convergence rate of our proposed extended level set method we have chosen a circular density jump which is passively advected within a

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parallel flow. The unit square domain is resolved by 32 x 32, 64 x 64 and 128 x 128 grid cells. Periodic boundaries are imposed. The LI error GI of the density is measured taking the difference of a coarse grid and fine grid solution according to :

rih is the number of grid cells of the fine grid in each direction, p2/l the density of the coarse solution and ph the density of the fine grid solution. The rate of convergence p1 based upon two L\ error approximations eij 64,32 and ei) 128,64 is given by :

The initial data are : radius 0.15, centered at (x0,yo) — (0.5,0.5), density ratio 1000:1, UOQ = 0.25, t>oo = 0.25. When integrating from t=0.0 to t=0.5 a convergence rate of 1.8 was achieved.

5.2 Rising bubble The governing equations for the considered variable density zero Mach number regime are (using the pressure decomposition p = PQ + M2£/2\ where -J^PQ = 0, since the flow is considered as incompressible):

H denotes a smoothed Heavyside function. The numerical method for zero Mach Number variable density flow is described in [Sch 98]. A rectangular domain with a height to width ratio of 2 was chosen consisting of 128 x 256 grid cells. At the boundaries slip conditions were set. This problem is characterized

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by the density ratio of liquid and gas C\, the diameter to width ratio C^, the viscosity ratio C3, the Reynolds number Re, the Froude number Fr and the Weber number W. The characteristic number and their definitions are listed below:

Ci C2 C3

Pi / Pg D/LX Vl/Vg

714 0.25 6667

Re Fr W

pi Uref D / pi

D2/a

9.7 0.78 7.6

Surface tension is implemented in the same way as explained in [Sus 94]. The initial shape of the interface is circular. The interface (thick line), streamlines (thin lines) and velocity arrows are plotted for times £=0.0, 1.0, 2.0, 3.0 below. In figure (3) the relative mass error (mass according to the distribution of G) is plotted versus time.

Figure 2: Interface (thick line), streamlines (thin lines) and velocity arrows at times £=0.0, 1.0, 2.0, 3.0 6. Concluding Remarks

The present scheme for fluid interface tracking methods avoids mass losses by incorporating a finite volume mass conservation equation. The inconsistencies between level set transport and mass conservation are eliminated by controlling volume fraction deviations through a suitable "penalty" - method

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Figure 3: Relative mass error resulting from the distribution of G versus time applied to the level set transport. Unphysical intermediate densities in nonmixed cells which can arise due to the capturing features of the mass conservation discretization are removed by a mass redistribution step. In our present approach the thickness of the intercase is zero. Artifical smearing of densities is neither necessary nor does it occur during a computation. Acknowlegdments

This work is supported by the Deutsche Forschungsgmeinschaft through project KL-611/5 - 1,2, in the framework of the CNRS-DFG Programme "Numerische Stromungssimulation" . Helpful discussions with Dr. Anne Boulioux (Montreal) and Dr. Raz Kupferman (Jerusalem) are appreciated.

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Bibliography

[Ada 99]

Adalsteinsson, D. und Sethian, J. A., The Fast Construction of Extension Velocities in Level Set Methods, Journal of Computational Physics, 148, 2-22, 1999

[Bou 95]

Bourlioux, A., A Coupled Level-Set Volume-Of-Fluid Algorithm for Tracking Material Interfaces, 3rd Annual Conference of the CFD Society of Canada, June 25-27, 1995, Banff, Alberta, Canada

[Cha 96]

Chang, Y. C., Hou, T. Y., Merriman, B. und Osher, S., A level Set Formulation of Eulerian Interface Capturing Methods fo Incompressible Fluid Flows, Journal of Computational Physics, 124, 449-464, 1996

[Puc 97]

Puckett, E. G., Almgren, A. S., Bell, J. B., Marcus, D. L. and Rider, W. J.,A High-Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows, Journal of Computational Physics, 130, 269-282, 1997

[Rid 95]

Rider, W. J., Kothe, D. B., Mosso, S. J., Cerutti, J.H. and Hochstein, J.I. Accurate Solution Algorithms for Incompressible Flows, 33rd Aerospace Sciences Meeting and Exhibit, January 9-12, 1995, Reno, NV

[Rid 95/2]

Rider and W. J., Kothe, Stretching and Tearing Interface Tracking Methods, 12th AIAA CFD Conference, June 20, 1995, San Diego. Paper number AIAA-95-1717

[Rid 98]

Rider, W. J. and Kothe, D. B., Reconstructing Volume Tracking, Journal of Computational Physics, 141, 112-152, 1998

[Sch 98]

Schneider, T, Botta, N, Geratz, K. J. and Klein, R., Extension of finite volume compressible flow solvers to multidimensional, variable density zero Mach number flow, submitted to the Journal of Computational Physics, preprint available at http://www.zib.de/thomas.Schneider/pub.html

[Set 96]

Sethian, J. A., Level Set Methods, Cambridge University Press, 1996

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[Sus 94]

Sussman, M., Smereka, P. und Osher, S., A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow, Journal of Computational Physics, 114, 146-159, 1994

[Sus 99]

Sussman, M., Almgren, A. S., Bell, J., Colella, P., Howell, L. H. und Welcome, M., An Adaptive Level Set Approach for Incompressible Two-Phase Flows, Journal of Computational Physics, 148, 81-124, 1999

Coupling mixed and finite volume discretizations of convection-diffusion-reaction equations on nonmatching grids

Raytcho D. Lazarov, Joseph E. Pasciak, and Panayot S. Vassilevski Department of Mathematics, Texas A&M University, College Station, TX 77843 and Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.A.

ABSTRACT In this paper, we consider approximation of a second order convectiondiffusion problem by coupled mixed and finite volume methods. Namely, the domain is partitioned into two subdomains, and in one of them we apply the mixed finite element method while on the other subdomain we use the finite volume element approximation. We prove the stability of this discretization and derive an error estimate. Key Words: combined mixed and finite volume methods, non-matching grids.

1. Introduction Coupling different numerical methods applied to different parts of the domain of interest is becoming an important tool in numerical analysis, scientific computing, and engineering simulations. In the coupling process several important mathematical issues arise that have to be addressed. First problem is to find what natural and stable mathematical formulation will lead to a good computational scheme. In the case of different methods used in different parts of the domain this means to find a stable way of gluing together the solutions in the subdomains. Secondly, we have to find an approximation of the mathematical formulation which is stable, convergent, and accurate. And finally, we have to construct and study efficient solution methods for the resulting algebraic problem.

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We shall consider the following homogeneous Dirichlet boundary value problem for the convection-diffusion-reaction equation:

where C — £Q + C, Cop = — V • aVp is the diffusion operator, and Cp = V • (pb] + CQP is the convection-reaction operator. Here Q is a bounded polygon in 7£d, d = 2,3 with a boundary <9fi, a = a(x] = {a^j(x)} is a d x d symmetric and uniformly in fJ positive definite matrix, and / = f(x) is a known function in L 2 (fi). Also b = b(x) = (bi, • • •, bj) is a given vector field and c0 = CQ(X) is a given function. We assume that b_(x] and CQ(X) are uniformly bounded in fi and satisfy the condition

This in turn guarantees the coercivity of the operator C in L 2 (f2) and the existence and uniqueness of its solution in the Sobolev space H01(^,}. This problem is a prototype of mathematical models in heat and mass transfer, diffusion-reaction processes, flow and transport in porous media, etc. In this paper we propose and study numerical methods for this problem when in different parts of the domain different discretizations on independent meshes are used. Namely, we consider mixed finite element approximation in one part of the domain and finite volume element method in the rest of the domain. It is important to note that coupling mixed finite element and finite volume or Galerkin finite element approximations does not require any auxiliary (mortar) space on the interface of the subdomains. This is due to the fact that the Dirichlet boundary conditions are natural for the mixed formulation, while the Neumann boundary conditions are natural for the standard weak formulation of a second order elliptic problem. In the recent years there has been growing interest in the finite volume method (called also control-volume method or box-schemes). This interest is mostly due to the requirement of many applications of having locally conservative discretizations. This is a discrete variant of the property of the continuous model which expresses conservation of certain quantity (mass, heat, momentum, etc) over each infinitesimal volume. The finite volume method has been combined with the technique of the finite element method in a new development which is capable of producing accurate approximations on general simplicial and quadrilateral grids (see, e.g. [4, 5, 6, 7, 8, 13]). For a collection of theoretical results and various applications we refer [2]. The main advantages of the finite volume method are compactness of the discretization stencil, good accuracy, and discrete local conservation, which for many applications is a very

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desirable feature of the approximation. Also, this method has well developed approximation schemes for convection and convection-dominated problems. The structure of the paper is the follow. In Section 2 we introduce all necessary notations and the weak form of the problem (1) in a two domains setting. In the subdomain where the mixed formulation is used we apply the more general concept of the discontinuous Galerkin method for second order equations in mixed form. In the case of convection-diffusion problems the pressure should be smoother than just L2 so we use the space H]oc- Further, we study the stability and derive an a priori estimate for the solution. Section 3 is the central part of the paper. Here we introduce and study the coupling of the mixed finite element and the finite volume element method. Further, in Subsection 3.2 we discuss the coupled mixed and finite volume approximation of convection-diffusion-reaction equations. Finally, in Subsection 3.3 we prove the unconditional stability of the discrete scheme and derive an estimate for the error. 2. Variational formulation In this section we first introduce all necessary notations for splitting the domain of the problem (1) in two subdomains 17 = $7i U 1^2 and using two different formulations in each subdomain. The weak mixed formulation in QI is derived when the pressure p is in the space jf//oc(17i). In ^2 we use pressure space Hl(ft,2). We prove that the coupled mixed/Galerkin formulation is stable and derive an a priori estimate which is the prototype of estimates for the approximations schemes established further in the paper. 2.1. Two-sub domain coupled formulation We partition ill into two subdomains with an interface boundary F, i.e. 17 = 17i U F U 172 (see Figure 1) and use the standard notations for Sobolev spaces of functions defined on 17i and 172: H(div, 17i), L2(17j), i =• 1,2 and #o(172, <9172 \ F). Here the last space denotes the functions defined on 172 having generalized derivatives in L 2 ((7 2 ) and vanishing on 5172 \ F. The inner products in these spaces are denoted correspondingly by

and

by II

, which in turn define norms denoted

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Figure 1: Domain partitioning: £7 = HI U F U Q2-

Whenever possible we skip the subscript in L2-norms. The dual space of HQ (fJ 2 , 90,2 \ F) is denoted by H~l(£l-2), the space of the traces of the normal component of the vector-functions in H(div, f ^ i ) is denoted by ff~ 1 / / 2 (F), and the space of the traces of functions in #o(Q 2 , <9n2 \ F) is denoted by 1 /2 HOQ (F). The trace spaces are equipped with the standard Sobolev norms. Finally, we shall use the notation < •, • >r for the duality pairing between HQQ (F) and H~l/2(T). Further, we denote by n; the unit vector normal to dtli for i = 1,2 and pointing outward to the domain. Finally, we split the interface boundary F = F_ U F+ where F_ — {x € F : b(x) -HI < 0} and F+ = {x £ F : b(x] -HI > 0}. Note, that this splitting is with respect to the vector m. An illustration of these notations in 2-D is given on Figure 1. In QI we use a mixed setting of the problem (1). That is, we introduce the new (vector) variable u = — aVp. To distinguish the solutions in the subdomains we denote by pi = p\Qi and p2 = Pln 2 - The composite model will impose different smoothness requirements on the components p1 and p2More specifically, we will require that u € #(div, fii), p\ 6 L 2 (fii), and P2 G .H0(f2 2 , dft \ F). Note that p2 is required to vanish on <9£)2 \ F. Testing the equation a-1u + Vpi = 0 by a function v e H(div, 17), using integration by parts, the zero boundary conditions for p\ on d£l\ \ F, and the fact the trace of p\ on F is the same for the trace of p2 on F, one ends up with the equation,

Further, in order to describe the weak form of the equation

we need to allow discontinuous functions pi from the space

The functions in Hfoc(tli} have traces from both sides of the interfaces of the subdomains K. Namely, for a given function pi E Hioc(fli} we denote these traces by p° and p\, where "o" stands for the outward (with respect to K} trace and respectively, "i" stands for the interior trace. Next, we give the weak form of the above equation. We borrow this formulation from the discontinuous Galerkin methods (see, e.g. [10], pp. 189-196) by testing the equation by a function w\ G Hfoc(Qi). We note, that this setting is quite similar to the mixed finite element method for convection-dominated convection-diffusion-reaction equations (see, e.g. [12]). Since the functions from HIOC(£II) are piece-wise smooth with respect to the partition JC we shall integrate over each K 6 /C and then sum the results. Following [10] we find first the contributions of the advection-reaction operator Cp\ by introducing the bilinear form CK(PI, MI) for any subdomain K € JC:

Here n is the outer unit normal vector to dK. Next, we integrate by parts in each subdomain K and sum over all K € 1C. Thus, for pi, w\ G Hloc(£l\) we get:

Note that this bilinear from is well defined for both continuous and discontinuous functions with respect to the partition /C. From this expression we see that if the subdomain K has a side/face on F_ then the trace p° should be replaced by its counterpart from f7 2 , namely by Pi(x}. Also on F_ we have w{ — w\ and on <9£7i_ \ F_ we take p° = 0. Further, for a given function t(x) we denote by £_ = min(0, t) and t+ = max(0, t). Thus, we get the following weak form of the second equation valid for all w\ 6 ///oc(£)i):

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where

and

Finally, testing the equation (1) by a function W2 € £[$(0,2] d^t-2 \F), using integration by parts, the zero boundary condition for w^ on 5fJ2 \ F, and the fact that u • ni = — aVpi • ni = aVp2 • 112 on F, one arrives at,

for all There are various ways one can take into account the influence of the problem in the domain fii on the problem in D2. One of the possibilities, which we shall use further, is to try to make a formulation, which is stable for small diffusion coefficient (or even for vanishing diffusion). In this case it is very important to formulate correctly the boundary conditions. Namely, at the "inflow" part of the interior boundary the solution should be specified from the "outside" data. Taking into account that F+ is the "inflow" part of F for the subdomain 02, we add fr Piw? b-112 ds and subtract its equal Jr p^w? b-n.2 ds since on F we have pi = pi. Thus, we get the following form of the last equation:

for all

where

Thus, the coupled system for the three unknowns u e H(div, fix), p\ 6 H\oc($l\) and p2 G #o(02; <902 \ F) consists of the equations (3), (5), and (9)

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summarized as:

for all v E H(div, fti), wi e Hloc($li], and w2 G #0(^2; 5H 2 \r), respectively. The bilinear forms %•(-, •) are defined by (6), (7), (10), and (11), respectively. 2.2. Well-posed-ness of the composite problem Here we verify the existence and uniqueness of the solution of problem (12) and its stability in an appropriate norm. For this we shall need some additional notations. Let £, = {e} be the set of edges/faces of the subdomain QI from /C and £0 the set of interior for fii edges/faces. Recall, that HI and n2 are the outward unit normal vectors to HI and J7 2 , respectively. For any edge e G £Q denote by ne a fixed unit vector normal to e and let K+ and K~ be the two adjacent to e subdomains from the partition /C. For edges/faces that are on dfli we shall always assume that ne = ni. Further, denote by [vi] and v\ the jump and the average of the discontinuity of v\, respectively, along any edge e. More precisely, this is the difference and the arithmetic mean of the traces VI\K+ and VI\K- taken from both sides of e:

Further, we use the following natural norm for

All terms in the expression on the right are nonnegative and this defines a norm on the space Hloc($li] x ./^(f^; <9fJ2 \ H- Note, that under certain conditions on the vector field b this is a norm even if 70 =0.

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The stability of the composed problem (12) is based on the following theorem: Theorem 1 The solution of the problem (12) satisfies the a priori estimate:

where the *-norm is defined by (13). The proof of the above theorem is based on the following lemmas. Lemma 1 The bilinear form (6) defined for v\, w\ G #/ oc (fJi) can be transformed to the following form:

Furthermore,

Lemma 2 For all

3. Coupling mixed and finite volume approximations of the convectiondiffusion equation Our approximation strategy is based on the finite volume method in the framework studied by Cai [4], Cai, Mandel, and McCormick [6] and also by Bank and Rose [1].

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3.1. An outline of the finite volume element method We first outline a finite volume discretization method for the case of pure diffusion problem posed on fi2,

Here / £ L2(^2) is given and TJN, for the time being is assumed given in the space L2(T). The finite volume method under consideration uses two different finite dimensional spaces: a solution space W2 and the test space W% • The Wi is the standard conforming space of piecewise linear functions over the triangulation Ti = {T} of f)2 into triangles in 2-D and tetrahedra in 3-D (we call them simplices). To introduce the test space W% we need a dual partition V2 of the domain into the finite (control) volumes V. Let M denote the set of all vertices (nodes) of the triangles/tetrahedra from ?2 and let A/o be a subset of those vertices that are not on the Dirichlet part of the boundary <9H2 \ F. In each simplex T £ ?2 one selects an interior node XT- Next, in 2-D one links XT with the midpoints of the sides of the triangle. In this way the triangle is split into three quadrilaterals. In 2-D, one can select XT to be the orthocenter of the finite element T and then the edges of the volume V(x) will be the perpendicular bisectors of the finite element edges (see the right Figure 2). With each vertex x G A/" of a simplex from ?2, we associate a volume V = V(x) that consists of all quadrilateral/polyhedra having x as a vertex (see Figure 2 for finite volumes in 2-D). The splitting of £1% into finite volumes V forms the partition V2 (see, Figure 3). Consider now the test space W£ spanned by the characteristic functions of the volumes V G V2 and that vanish at the nodes M \ A/o on the boundary d^l-2 \ F. If one defines the piecewise constant interpolant I£ with respect to the volumes V G V2, then the space W% is actually equal to I^W-i because they have the same degrees of freedom (associated with the vertices x € A/")-

The L 2 (ft 2 ) and Hl(ft,2) norms in W2 are defined in a standard way. We shall need also discrete variants of these norms for functions in W2*. First, we define the interpolation operator //> : W£ i—> Wh by the following natural rule: Ih^2 is ^ne piece-wise linear interpolant of v2 over each finite element T G ?2Then we define H^lli./i = 11-^2 111,n 2 - This norm is essentially formed by the squared differences of the values of v% at the vertices of each finite element.

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Figure 2: A finite volume V associated with a vertex from the primal triangulation. Left: the vertices of V interior to the triangles T are arbitrary, whereas those on the edges of T are midpoints.

Figure 3: In HI we use the lowest order Raviart-Thomas spaces over the finite elements T; in ^2 we use a solution space W% of continuous functions that are linear over the finite elements T and a test space W£ of pice-wise constant functions over the volumes V.

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Next, we define the following, in general nonsymmetric, bilinear form on W2 x W%:

Then the finite volume approximation of (15) is: find v% G W^ which satisfies the following identity for all w\ G W2*;

We note that the integrals over dV for V = V(x) a volume corresponding to a Neumann node x (i.e., if a: € F), contain only the interior (to f^) part of dV. We assume that the triangulation TI is aligned with the possible jumps of the coefficient matrix a(x), i.e. over each finite element T G ?2 the matrix a(x) has smooth elements. Therefore, there is a constant CQ > 0 such that for all

where These inequalities of two d x d matrices with real elements are understood in the sense of inequalities for the corresponding bilinear forms, i.e. a > a(x), iff £Ta£ > £Ta(:r)£, V £ G 7ld. Also, the above equality of the matrices 0(2;) and d(x) is understood in element-by-element sense, i.e. the elements of a(x] are the mean values over T of the corresponding elements of a(x). Obviously, in case of piece-wise constant coefficients a(x) = a(x] and CQ = 0. The well posed-ness of the finite volume element approximation follows from the weak coercivity of the bilinear from 02,71(^2,^2) f°r sufficiently fine partitions ?2. We have: Lemma 3 Let the partition Ti be so fine that h < I/Co, where the constant CQ is determined in (17). Then the following inequality holds true

with a constant C independent of h.

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In fii we use the lowest order Raviart-Thomas spaces. Thus, V as a subspace of H(div; fii), and W\ is the space of piece-wise constant functions with respect to the partition 71 and therefore a subspace of Hioc(Q,i) for /C = 71. Thus, the advection term Cp\ will be discretized using the pressure space W\ of piecewise constant functions on the triangulation 71. For the discretization in £1-2 we will use the space Wz C #o(fJ 2 , <9^2 \ F) of continuous piecewise linear functions on 72• Further, in the finite volume setting we use as a test space W% of piecewise constant functions on V2. Thus, applying equation (4) consecutively for (ui, w\) € W\ x W2, and (i> 2 ,u> 2 ) € W2 x W£, respectively, we get the mixed finite element and finite volume approximations, respectively, of the bilinear form corresponding to the first order term. However, like in the standard Galerkin finite element method this approximation of the operator C will lead to central differencing, which in turn will lead to a conditionally stable (only for sufficiently small step-size h) scheme. In order to derive a unconditionally stable scheme we shall use upwind approximation in 02. 3.2. Derivation of the coupled method

Since both v\ and wi are discontinuous piece-wise constant functions with respect to the triangulation 7i the formula (4) is applied in a straightforward manner for JC — T\ so we get the following approximations a^ and a^2 of the forms an and 012, respectively:

and

Now we find the contributions of the the operator C from 02 and we define the approximations of the bilinear forms a2i and a 22 . We shall simply rewrite

Since the functions in W^ are continuous then C(i; 2 ,w 2 ) is well defined for all V2 € W-z and u>2 G W£. Taking into account that the functions in W£ are

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Figure 4: The shaded area is the volume V centered at the vertex xl; the doted boundary of V denotes the "inflow" boundary, while the solid one is the "outflow" part; on the various pieces of the boundary dV we have the following approximation of the convection term: on (a, 6), (m,c), ( / , g ] and (g, a): v2(x) - v 2 ( x l ) ; on (6,m) : v2(x] - v2(x°] = v2(C); on (c,d) : v 2 ( x ) = v2(x°) - v2(D}] on (d,e) : v2(x) = v2(x°) - v 2 ( E ) ; on (e,f) : v2(x] = v2(x°)=v2(F). constant over each finite volume V 6 V2 then the contributions from each finite volume V G V2 are:

Since v2 is continuous then obviously, we have v2 = v2 = v2(x). On the boundary F+ the values v2 are not defined (this is the inflow boundary for Q2) and we shall take them from the corresponding counterpart in fii, i.e. as vi(x). Thus, we split the integrals over dV into two parts and get

Unfortunately, the exact calculation of the first integral in (21) will lead to central differences and therefore to a scheme which is stable only for sufficiently small step-size h. The limitation of the step-size h will depend on the magnitude of the convection coefficient 6 relative to the diffusion coefficient (matrix) a. For problems with dominating convection this will lead to prohibitively small h. In order to avoid this conditional stability we introduce an up-wind approximation of the integrals. This approximation is done in the following way. We denote by

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V(xl) a finite volume centered at the vertex xl and by V(x°) any of neighboring volumes centered at the vertices x°. The integral over dV \F_ is split into subintegrals over the boundaries of V 7 = V(xi] D V(x°) n T with its neighboring finite (control) volumes and contained in the finite element T. We assume that over each 7 the function b(x) • n does not change sign (i.e. is either nonnegative or negative). Then on 7 we use upwind approximation of the following form (for a 2-D illustration, see Figure 4):

Note, that in the finite volume V(xl) we have I^V2 (x) = v^x1}. Similar equalities are valid for the neighboring volumes V(x°] as well. Thus, roughly speaking the function v?,(x) has been replaced by its interpolant in the space of discontinuous functions W£ and then taken the appropriate (up-wind or in the opposit direction of the vector-field &(#)) values at the finite volume interfaces. A particular finite volume in 2-D is shown on Figure 4. Summing for all V € V^ we finally get the following form by taking also into account the diffusion term (16):

for all

and the form

The coupled mixed finite element/finite volume approximation of the composite problem (12) reads as: find u/! e V, pi^ 6 W\, and p2,h € W%, such that

for all

, and

respectively.

3.3. Stability of the coupled scheme and error estimate

An important feature of the described above discretizations is that the corresponding operator is coercive in an appropriate norm and the method is stable.

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For proving the stability we shall follow the same argument as in the case of original setting. Let as before £ = {e} be the set of edges/faces of the elements from 7i and let £0 be the set of interior edges/faces. Similarly, Q — {7} is the set of edges/faces, with each 7 being the boundary of two adjacent volumes Vi G V<2 and V2 G V^ contained in a finite element T, i.e., 7 = V\ n V-2 n T. This splitting can be also used in the computational procedure, since it will lead to element-wise contributions of the convection term to the stiffness matrix. Note, that all edges/faces 7 are in the interior of f^- For the coercivity of the coupled problem we need the following discrete variant of the norm (13):

Here a is a piece-wise constant matrix with respect to the partition ?2 defined

by (17). Theorem 2 Let h < I/Co, where C0 is defined by (17). Then the solution of the problem (12) satisfies the a priori estimate:

where the (*,h)-norm is defined by (25) with respect to the partitions 71 and T 2

Proof: As in the continuous case, by testing (24) with and w-2 — —Ifrp2,h we get the equation:

Further, the estimate (14) is a consequence of the simplified form (28) of o^ and (29) of a^, which are established in the lemmas below. Lemma 4 For any edge/face e denote by ne a fixed unit vector normal to e and let T+ and T~ be the two adjacent elements to e. Similarly, for any

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edge/face 7 £ Q denote by n7 a unit vector normal to 7 pointing to one of the neighboring volumes Vj~ and V~. Also, [v\\ denotes the jump of a function across an underlined boundary (here we have either e or j) and w\ denotes the arithmetic mean of the jump (introduced in Section ). Then,

Similarly, for all v% € W-z, w% £ W% the following identity is valid (to the expressions we have used the notation v^ = l^i)'-

simplify

Proof: The proof of (28) and (29) essentially repeats the arguments of Lemma 1. There is a small difference in the proof of (29) where the integrals over each 7 6 G have been computed by using up-wind approximation. Lemma 5 The following identity is valid for all

Similarly, for all

W<2 (here in order to simplify we use the notation

Finally, we have the following error estimate:

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Theorem 3 Assume that the solution p(x] of the problem (1) is H2 -regular in 0. Then the solution (u.h,P\,h,Pi,h) of the coupled mixed discontinuous finite element and finite volume methods (24) converges to the solution (u,pi,p2) of the composite problem (12) and the following error estimate holds true:

The constant C does not depend on h but may depend on the ratios

and

4. Bibliography [1]

R.E. BANK AND D.J. ROSE, Some error estimates for the box method, SIAM J. Numer. Anal. 24(1987), 777—787.

[2]

F. BENKHALDOUN AND R. VILSMEIER (Eos), Proc. First Intern. Symposium on Finite Volumes for Complex Applications, July 15 18, 1996, Rouen, France, Hermes, 1996.

[3]

F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991.

[4]

Z. CAI, On the finite volume element method, Numer. Math., 58 (1991) 713-735.

[5]

Z. CAI, J.J. JONES, S.F. McCoRMicK, AND T.F. RUSSELL, Control-volume mixed finite element methods, Computational Geosciences, 1 (1997) 289-315.

[6]

Z. CAI, J. MANDEL, AND S.F. MCCORMICK, The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal. 28 (1991), 392-402.

[7]

S.H. CHOU AND P.S. VASSILEVSKI, A general mixed co-volume framework for constructing conservative schemes for elliptic problems, Math. Comp., 68 (1999).

[8]

S.H. CHOU, D.Y. KWAK, AND P.S. VASSILEVSKI, Mixed covolume methods for elliptic problems on triangular grids, SIAM J. Numer. Anal. 35 (1998), 1850-1861.

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[9]

T. IKED A, Maximum Principle in Finite Element Models for Convection-Diffusion Phenomena, Lecture Notes in Numer. Appl. Anal., Vol. 4, North-Holland, Amsterdam New York Oxford, 1983.

[10]

C. JOHNSON, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987.

[11]

R.D. LAZAROV, J.E. PASCIAK, AND P.S. VASSILEVSKI, Iterative solution of a combined mixed and standard Galerkin discretization methods for elliptic problems, (submitted to Numer. Linear Alg. Appl.).

[12]

M. Liu, J. WANG, AND N.N. YAN, New error estimates for approximate solutions of convection-diffusion problems by mixed and discontinuous Galerkin methods, Univ. of Wyoming, Preprint, 1997.

[13]

I.D. MISHEV, Finite volume methods on Voronoi meshes, Numerical Methods for Partial Differential Equations, 14 (1998), 193-212.

[14]

P.A. RAVIART AND J.M. THOMAS, A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods (I. Galligani and E. Magenes, eds), Lecture Notes in Math., Springer-Verlag, 606 (1977), 292-315.

[15]

P.A. RAVIART AND J.M. THOMAS, Primal hybrid finite element methods for second order elliptic equations, Math. Comp., 31 (1977), 391-396.

[16]

M. TABATA, A finite element approximation corresponding to the upwind finite differencing, Mem. Numer. Math., 4 (1977) 47-63.

[17]

C. WIENERS AND B. WOHLMUTH, Coupling of mixed and conforming finite element discretizations, American Mathematical Society, Cont. Math. 218 (1998), 547-554.

Numerical computation of 3D two phase flow by finite volumes methods using flux schemes

J.-M. Ghidaglia Centre de Mathematiques et de Leurs Applications ENS de Cachan and CNRS UMR 8635 61 avenue du President Wilson 94235 Cachan Cedex France jmgQcmla.ens-cachan.fr http://www.cmla.ens-cachan.fr/~jmg

ABSTRACT We propose here a general class of cell centered finite volume methods specially designed for the discretization of complex models of partial differential equations like those occuring in 3D two phase flow. After a brief introduction to these models (the so called averaged models in Eulenan formulation), we develop all the tools which are needed in order to arrive to a fully discrete scheme suitable for coding. Hence we discuss conservative systems, non conservative ones, time discretization, discretization of source terms, of diffusion operators, of boundary conditions, .. . We also briefly discuss non conformal meshes. We strongly rely on the concept of flux scheme which is, according to us, very well suited for the problems considered here. Key Words : Two phase flow, Flux schemes, Finite volumes, non hyperbolic convection, Source terms, Footbridges.

1. Introduction

Our goal in this contribution is to discuss a class of cell center finite volume methods, on unstructured conformal or non conformal meshes, designed for complex models (such a goal was also addressed by T. Gallouet [GAL 96]

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in the previous version of this symposium). We are mainly motivated by the computation of two phase flows occuring in several fields of the nuclear power industry (of course there are many other contexts where multiphase models occur), that is flow of water and its vapor. Models pertaining to this field encompass many difficulties, whose non exhaustive list is follows : (i) the solutions might develop strong gradients, (ii) the equations might involve stiff source terms, (iii) non conservative products might be present in the equations, (iv) the thermodynamics could be highly complex (e.g. phase transition), (v) the total convection operator might be non hyperbolic, (vi) in some part of the flow low Mach numbers can occur,

(vii) . . . (viii) and last but not least : there is no "universal model" to work with. Some of these features are present in one phase Computational Fluid Dynamics (Iph-CFD), other are typical in combustion but 2ph-CFD often involves toghether all of them. In this paper we shall propose a very general framework for the discretization of such systems. Generality is essential in this context according to (viii). And passing from one model to an other (which might implies even the variation of the number of scalar equations to be considered) should cause only minor changes in the code. Moreover one should ask almost no a priori conditions on the model (besides the fact that it is build upon physical considerations). One of the major by-product of this stategy is that a code derived according to this philosophy can be a tool for the validation of the physical models, and this point is one of the main issues in 2ph-CFD at the present time. Most of the results presented here were inspired by a very fruitful colaboration with the Departement Transfert Thermique et Aerodynamique (TTA/D R&D/EDF) and with the Service de Simulation des Systemes Complexes et de Logiciels (SYSCO/DRN/CEA).

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2. A short discussion of the models

We consider the motion of two non miscible fluids, like e.g. air and water or water and steam. Since we are concerned with macroscopic dynamics, we shall consider a continuum model. For single phase flows, except for very hight speed rarefied flows, Navier-Stokes Equations provide a hightly satisfactory continuous model. For two phase flows, a first approach could consists in writting for each phase a Navier Stokes model completed with jump relations across interfaces between the two fluids. This is of course possible for very very simple situations and cannot be envisaged in general. We shall consider here so called "two fluid" models (in Eulerian formulation) obtained throught an ad hoc averaging process. That is at each point in space and time, the modelization assumes that both fluids are present and a parameter (which is also to be determined) represents the proportion of each fluid. Such a model can be derived by an averaging process and we refer to e.g. Ishii [ISH 75], Ransom [RAN 89], Drew and Wallis [DRE 94] for more details. The models rely on the three usual balance equations (mass, momentum and energy) for each fluid. Denoting with subscripts v and / the quantities attached to one of the two fluids (e.g. density, velocity, energy,..) and by av, a/ the volume fraction of each fluid : av -f a/ = 1, these equations reads as follows (*£{»,/}). Balance of mass.

where p^ is the density of the fluid k, u^. the velocity of the fluid k and F^ the density of the mass transfert to the fluid k resulting from interfacial exchanges with F7; + T; = 0. Balance of momentum.

where P is the pressure in the mixture, Tk the viscous stress tensor, g the gravity field, Ik the volumic density of the momentum transfert to the fluid k resulting from interfacial exchanges (after substraction of the pressure contri-

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bution) and Balance of total energy.

where e^ is the specific internal energy of the fluid k, hk the specific enthalpy of the fluid k, &kQk the volumic heat transfered to the fluid k, QkQk the heat flux in the bulk of the fluid k, Hk the volumic density of the energy transfered to the fluid k resulting from interfacial exchanges with Hv + H/ = 0. Once this set of differential equations written, it remains to close the system by adding enought closure relations in order to obtain the same number of equations and independent unknowns. At the present time, as emphasized in the Introduction, it is an open problem. However by using various methods it is indeed possible to obtain, in definite physical contexts, some closed models from the equations above. These models will in general leave free a few number of constants that are to be determined by experiments (correlations). Hence the numerical modelisation of two fluid flows must be strongly coupled with experimental programs. Basic models We give below three typical models (that can be written in 1, 2 or 3 space dimensions). These models are very simple in the sense that they do not involve complicated physical correlations in their right hand side. Actually these models are used in the context of numerical benchmarks designed in order to check that the numerical method used is able to capture some of the important features of the flow considered and also in order to evaluate the numerical dissipation of the method. We refer to [MIM 99] for a collection of numerical benchmarks.

2.1. A 3 equation model This is the so called homogeneous 3 equation model that we write for the sake of simplicity in ID :

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Here p represent the density, u the average velocity, ur the relative velocity between phases, c the quality, L the latent heat and 3> a heat flux (source term). See also Example 1.

2.2. A 4 equation model This two-fluid model reads as :

Here p^ represents the density of the phase fc, u^ its velocity, p is the pressure, and a/5 denotes the volume fraction of the phase k, (av + at = I). The relative velocity is simply ur — uv — u^. Here, the source term is a drag force which is given by the correlation k = ^avpi^- in equations (9) and (10), where R^ is the bubble radius and CD the drag coefficient. 2.3. A 6 equation model In the previous two fluid model, no energy balance equations were written. This corresponds to isentropic flows. Here we want to consider the full set of balance equations (mass, momentum and energy) and this leads to a so called six equations model (which lead in 3D to a set of 10 scalar evolution partial differential equations). Let us consider the following system of equations :

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We still have to describe the thermodynamics of this "one-pressure" model. To this extent, we assume that two pressure laws are given, one for the "gas" and one for the "liquid" phase :

Here e^, denotes the internal phase energy and

3. The finite volume approach

The goal of this section is to describe the program that lead to the full discretization of the models of Section 2. We shall not present numerical results refering to the bibliography for that purpose : [ACT 99], [BOU 98], [GKL 96], [GKT 97], [GP1 99] , [HAL 98], [KUM 97], [PAS 99], [TAJ 96], [TAJ 98], [TKP 99]. In Section 3.1, we first address the conservative case which allows us to introduce the basics of the finite volume discretization. Then in Section 3.2, we introduce the characteristic flux approach. This lead us in Section 3.3 to the concept of flux schemes. In Section 3.4, we extend the previous analysis to non conservative systems while Section 3.5 deals with the time discretization. In Section 3.6 we put the emphasis on the discretization of forcing terms while Section 3.7 is devoted to the implementation of boundary conditions. In Section 3.8, we propose a convenient way of discretizing diffusion operators and finally in Section 3.9 we give some elements on the generalization to non conformal meshes.

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3.1. The conservative case Althought almost all the models encountered in two phase fluid flow are non conservative, we are going to deal first with conservative ones since they are convenient in order to introduce the basic notations. Let us consider a system of m balance equations :

where here Dosed m a nu-dimensional domain

maps

This equation is in practice).

Example 1 The homogeneous model (4) to (6) enters in this category. Assuming that a thermodynamic law p = P(p,e) is given, and when ur = 0, one recovers the usual Euler equations. In the case where ur ^- 0, we have also to prescribe the quality c = C(p, e) and in this case, we can write (4) to (6) into the form (18) provived we set v\ — p, v? = pu, vs = pE, F± = pu F} = p(u2 +c(l - c}ul] + p, F3! = p(uH + urc(l -c)(L + uur + (| - 2c)uj!)). Here E = e+||u| 2 , H = E-\-V-. It should be noticed that even when ur is a fixed constant and even for the physical laws of water, this system is not necessarily hyperbolic [TAJ 98]. We assume that the computational domain Q is decomposed into smaller volumes (the so called control volumes) K : Q, — UK^T^ and, in a first step, we assume that Q = U^gy-A' is "conformal" i.e. that it is a finite element triangulation of £1. In practice one can use triangles for nd = 2 and tetrahedrons for nd — 3. The cell-center finite volume approach for solving (18) consists in approximating the means

where vol(K) denotes the nd-dimensional volume of K and area(A) stands for the (nd — l)-dimensional volume of an hypersurface A Integrating (18) on K makes the normal fluxes, FQK, appear

where dK is the boundary of K, J'(cr) the unit external normal on dK and da denotes the (nd-l)-volume element on this hypersurface. Indeed, we have

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according to (18) :

The heart of the matter in finite volume methods consists in providing a formula for the normal fluxes FQK in terms of the {VL}L^T- Assuming that the control volumes K are polyhedra, as it is most often the case, the boundary dK is the union of hypersurfaces K D L where L belongs to the set A/"(/C), the set of L € T, L / /\, such that K fl L has positive (nd — l)-measure. We can therefore decompose the normal flux as a sum :

where (v,k.l points into L) :

Motivated by the case where (18) describes wave propagation phenomena, it is natural to look for an approximation of (23) in terms of vx(t) and v ^ ( t ) :

where $ is the numerical flux to be described. At this level of generality* must satisfy 2 structural properties : (i) consistency : $(w, w] K, L) = F(w) • VK,L, (ii) conservation: $(t;, w; K, L) = — &(w, v; L, K). 3.2. The characteristic flux approach ([GKL 96]) Let us consider the equation*

where v G Mm and / : Mm H-> Mm: (i.e. (18) ) when nd = 1). We denote by A(v) the jacobian matrix ^' and observe that since A(v}^ = ^y' then according to (25),

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This shows that the flux f(v) is "convected" by A(v) (like v since we also have

Let us assume that (25) is smoothly hyperbolic that is to say : for every v there exists a smooth basis ( r i ( v ) , . . ., rm(v)) of R m made with eigenvectors of A(v] : 3\k(v) G M such that A(v)rk(v) = \k(v)rk(v). It is then possible to construct ( l i ( v ) , . .. , l m ( v ) } such that t A(v}lk(v] = \ k ( v ) l k ( v ) and lk(v)-rp(v) — &k,PThe numerical flux <J> represents the flux at an interface. Using a mean value p of v at this interface, we replace (26) by the linearization :

It follows that setting fk(v) = Ik(l^) ' f ( v ) , which is termed as the k-th characteristic flux, we have

Solving exactly this last equation (see also Remark 7), leads to the following formula :

Remark 1 Non hyperbolic convection operators In the complete physical models, the equations for the balance of momentum and energy (see (2) and (3)) involve terms with second order gradients like e.g. molecular and turbulent dissipation and diffusion. Here we are only discussing the connective part, i.e. terms with first order gradients, see Section 3.8 concerning the discretization of second order gradients. It might happen (this is very often the case when dealing with two phase flow, [STE 84]), that this convection operator is not hyperbolic (see also Example I). That is the matrix A(v) might have complex eigenvalues. When it is the case, we keep formula (29) unchanged but this time, \k(v) denotes the real part of the k-th eigenvalue of A ( v } . The physical meaning of this is that we still identify these numbers as physically meaningful wave speeds.

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3.3. Flux schemes We emphasize the fact that formula (29) has the following remarkable property : the numerical flux at an interface <£ is a linear combination of the fluxes f ( v ) and f ( w ) on both sides of the interface. This has led to the definition of flux schemes ([GHI 98]). Definition 1 The numerical flux <$ corresponds to a flux scheme when there exists a matrix U(v,w,K,L) such that

The extension of formula (29) to the multidimensional case is therefore given throught the following definition. Definition 2 The numerical flux of the "VFFC" method is obtained by formula (30) when we take :

where n(v,w]K,L) is a mean between VK and VL which only depends on the geometry of K and L :

Remark 2 In formula (31) the sign of a matrix M is the matrix which has the same generalized eigenspaces as M but whose eigenvalues are the sign ( —1, 0 or +1) of the corresponding eigenvalues of M. When the eigenstructure of M is not explicitely known, this matrix can be numerically computed in two different ways, . First one can use a numerical package that determines this eigenstructure. Second one can use a very efficient iterative method (a variant of the Newton-Schultz algorithm, see [ALO 99]) which produces a polynomial sequence of matrices converging rapidly to the matrix sgn(M). Remark 3 Other flux schemes Many other numerical schemes are flux schemes (see [GHI 98]). They have in general a very similar behavior as far as numerical accuracy is concerned. For example Roe's scheme is a flux scheme. For complex models like those considered here, determining Roe's average can be a

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difficult challenge while our VFFC scheme involve a simple geometric average. We note also that Stddke et al [SFW 97] have proposed also a flux scheme (for a particular system) with a simple average. Concerning Roe's scheme, Toumi [TOU 92] has derived very efficient procedure in order to use this scheme in the context we consider here. Later Halaoua [HAL 98] has generalized Roe's scheme to non hyperbolic convection operators. We particularly recommend the paper by Toumi, Kumbaro and Paillere [TKP 99] for a very nice review on these questions (see also [GLT 99]. Proposition 1 Combining (21), (22), (24), (30), (31) and (32), the finite volume approximation of (18) is the following system of o.d.e. 's :

3.4. The non conservative case Let us now consider a system of m balance equations which reads as :

and which therefore generalizes (18). The terms Cj(v)j£- are called non conservative products, and they introduce new difficulties both from theoretical and practical point of view. Often, such terms come from modelisation (e.g. approximations, closure relations,...) procedures and according to the way they are dicretized, one can obtain non physical solutions or even numerical divergence phenomena. Such terms are present in almost all the models encountered in two phase flow fluid dynamics and they represent one of the major difficulty in this area. Example 2 The two fluid model (7) to (10) enters in this category when k = 0 (otherwise we just have a source term on the right hand side of (34))- Here we take v\ = avpv, v? = otipi, vs = avpvuv, v4 = atptut, F/ = avpvuv, F\ — ottptut, Fg = av(pvul -\-p), F| — ai(piu^ + p) and the matrix 6*1(7;) is such that :

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Remark 4 Of course the form 34 is not unique since one can add an arbitrary function of v, G, to F and substract VvG(v] to the matrix C(v). Since the function F which appears has a physical meaning (the so called physical flux), its form is chosen on physical ground rather than on an abstract mathematical principle. For example the choice made in Example 2 is dictated by the property that when we have only one phase which is present (e.g. when av = 0) one recovers the classical fluxes of the Euler equation. Actually, other non conservative products can occur, and the general system of equations reads as :

Example 3 For the 3-dimensional case (nd = 3), we can recast system (11) to (16) under the form (35) with m = W ; v\ = avpv, v^ — (*tpt, t>3 = avpvUVii, V4 =

ttlPl^,!,

^5 = C*vpvUVt2, VQ = QlpiU^^, ^7 = CtvPvUVi3, V8 =

Q

tPiut,3, VQ — avpvEv and VIQ — aipeEt. Then one gets the F1- like in Example 2 (they are given expilcitely in [SOU 98]). Assuming that the matrix I + D(v) is invertible (which is the case in practice), we can transform easily (35) into (34). Hence we can only discuss the discretization of this last equation. In the one dimensional case, equation (34) reads as :

Denoting this time by J ( v ) the jacobian matrix

where A(v) = J ( v } ( I -+- C ( v ) J ( v ) formula (31) is replaced by

and

1

we have instead of (26),

). Hence, in the multidimensional case,

where

and

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Remark 5 Of course when (7 = 0 both formulas (31) and (38) agree. Now that we have a formula for computing the numerical flux at an interface K D L, we have to discuss the discretisation of the non conservative products. Integrating equation (18) on a volume control K, leads to the question of approximating the integral fK Y^]=i Cj(v)-j£- dx. Following [BOU 99] (see also [BOU 98]), we take :

where EK,L is the following m x m matrix :

Proposition 2 In the non conservative case, the finite volume approximation of (34) ^s the following system of o.d.e. 's :

3.5. Fully discrete schemes In the previous Section, we arrived to systems of o.d.e.'s ((33 and (41)). In order to actually compute the solutions, we have to perform a time discretization. This discretization can be either explicit or implicit. As it is well understood, the explicit discretization of the convective flux will lead to a stable scheme only under a Courant-Friedrichs-Lewy (C.F.L.) condition. This condition will impose on the time step a very restrictive constraint. Namely that this number must be smaller than Ax/c where Ax is a measure of the smallest length in the space discretization and c a measure of the largest speed of propagation appearing in the physical problem (note that in liquid and vapor flows this speed can be of the order of 2000 ms~1}. Hence when there are long transients, explicit methods are not effective in this context.

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We consider (34). The previous Section was devoted to the space discretization of this equation and we arrived to (41). The Euler explicit scheme for this equation reads :

As already said, this scheme is stable only for small time step. An alternative is then Euler implicit schemes. We solve the following equation for v^+l :

Althought of interest, this method can be too diffusive since all the characteristics are made implicit regardless the magnitude of the characteristic speed. We have proposed in Ghidaglia, Kumbaro, Le Coq and Tajchman [GKT 97] an alternative method in ID (see Remark 7) which has been generalized to the nd-D case by Boucker [BOU 98]. Definition 3 In the multidimensional case, the numerical flux of the "implicitexplicit VFFC" scheme is given by

where the m x m matrix

{1, 2, 3, 4} are defined as follows :

with n denned bv

the eigenelements (l,r and X) refer to the jacobian matrix the uj£ are given according to the following cases. (i) When 0

[

and

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When

(iv)

When

(v)

When

83

Above, &K,L denotes a characteristic lenght in K which depends also on L and K is a nondimensional number between 0 and 1 to be specified (see Remark 6). Remark 6 We can take for example

and K is determined thanks to the stability limit of the explicit scheme, see Boucker [BOU 98] (for example K w 0.3 for nd = 3y). Remark 7 the one dimensional case When nd = 1, formulas (44) to (51) are changed as follows :

where the m x m matrix

{1, 2, 3, 4} are defined as follows :

with n is defined by

and uj^ according to the following conditions.

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(i) When 0

(ii) When

(m) When

(iv) When

(v) When

In the case where the C.F.L. conditions :

are satisfied, formula (29) follows from these considerations.

3.6. On the discretization of source terms

Instead of (25), we want to solve

by the numerical scheme :

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Equation (63) can be obtained by integration of (62) on the rectangle [tn, tn+i] x [xj_i/2, Xj+i/2\ and this leads to the following form for E™ :

Actually, this formula is not suitable when the source term is large and one must modify (64) according to the expression of the numerical flux 9, where and

where U(-,-} is a matrix to be specified. Due to the fact that the numerical flux is not centered (upwind bias) one can observe large errors on the permanent solution under investigation. Let us introduce the notion of enhanced consistency as follows. Denoting by

we say that E^ satisfies, with respect to the numerical flux g, the enhanced consistency property when we have : if at some time-step, vj is such that

then v? +1 given by (63) must be equal to v^. This condition can be equivalently formulated as

if u" satisfies (67). In Alouges, Ghidagliaand Tajchman [AGT 99], the following result is shown. Theorem 1 Let (a,-) be given in the form (65) and denote by Un,L the matrix 3 ' 2 Uj+i(v^, y ?+i)- The enhanced consistency (i.e. (68)) will be satisfied if we discretize the forcing term S according to the following formula

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Observe that from the computational point of view, formula (69) induces almost no extra cost since Uj+± has already been computed in order to construct the numerical flux. In order to illustrate formula (69), let us show what it means in the case of a single linear equation vt + c vx = S where e.g. c > 0 and v G M. Here the usual upwind scheme amounts to take [/" = 1, i.e. a(v. w] = c v and (69) reads as :

3.7. On the boundary conditions So far we have not discussed the implementation of boundary conditions. This is a very important matter since they actually determine the solution. Let us consider the space discretization of system (18) by our cell centered finite volume method. We have found the sytem of o.d.e.'s (33). Of course this evolution equation is not valid when K meets the boundary of Q. When this occurs, we have to find the numerical flux $(VK, A", d£l). In practice, this flux is not given by the physical boundary conditions and moreover, in general, (18) is an ill-posed problem if we try to impose either v or F(v) • v on d£l. This can be simply understood on the following linearisation of this system :

where v represents the direction of the external normal on iacobian matrix :

is the

and v_ is the state around which the linearisation is performed. When (18) is hyperbolic, the matrix J^ is diagonalizable on IR and by a change of coordinates, this system becomes an uncoupled set of m advection equations :

Here the c^ are the eigenvalues of J_^ and according to the sign of these numbers, waves are going either into the domain Q (c/c < 0) or out of the domain H (c/j > 0). Hence we expect that it is only possible to impose p conditions on

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K fl dSl where p = ${k e {1,. . ., m) such that ck < 0}. Let us consider now a control volume K which meets the boundary dQ. We take v_ — vK and write the previous linearisation. We denote by x the coordinate along the outer normal so that (71) reads :

which happens to be the linearisation of the ID (i.e. when nd = 1) system. First we label the eigenvalues Cf~ (v_) of J^ by increasing order :

(i) The case p — 0. In this case all the informations come from inside £1 and therefore we take :

In the Computational Fluid Dynamics litterature this is known as the "supersonic outflow" case. (ii) The case p = m. In this case all the informations come from outside fi and therefore we take :

where $given are the given physical boundary conditions. In the Computational Fluid Dynamics litterature this is known as the "supersonic inflow" case. (iii) The case 1 < p < m — I. As already discussed, we need p informations coming from outside of f2. Hence we assume that we have on physical ground p relations on the boundary :

Remark 8 The notation gi(v) — 0 means that we have a relation between the components o f v . However, m general, the function gi is not given explicitely in terms of v. For example gi(v) could be the pressure which is not, in general, one of the components o f v .

88

Finite volumes for complex applications Since we have to determine the m components of <&(VK, K, d£l), we need m — p supplementary informations. Let us write them as

In general (78) are termed as "physical boundary conditions" while (79) are termed as "numerical boundary conditions". Then we take : where v is solution to (78)-(79) (see however Remark 10 and (86)). Remark 9 The system (78)-(79) for the m unknowns v^Gisamxm nonlinear system of equations. We are going to study its solvability, see Theorem 2. Let us first discuss the numerical boundary conditions (79). By analogy with what we did on an interface between two control volumes K and L, we take (recall that v_ = VK) '•

In other words, we set hf,(v) = lk(vK)-(F(v)-i/K)—lk(vK)-(F(vK)-VK)We have denoted by ( / i ( v ) , . . . ,/ m (v)) a set of left eigenvectors of J^ : tJ_^lk(p_) — Cklk(v_) and by ( r i ( u ) , . . . , rm(v_)) a set of right eigenvectors of J^ : J^rk(v_) = c/ c r/ t (v). Morover the following normalization is taken : lk(v_) • rp(v_) — $k,pAccording to Ghidaglia [GHI 99] we have the following result on the solvability of (78)-(79). Theorem 2 Assume that

With the choice (81) the nonlinear system (78)-(79) has one and only one solution for v — v_ sufficiently small. Remark 10 In practice, (78)-(79) are written in a parametrized manner. We have a set of m physical variables w (e.g. pressure, densities, velocities,...) and we look for w satisfying :

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and then we take : The system (83)-(84)-(85)

3.8.

is then solved by Newton's method.

On the discretization of viscous fluxes

Our aim is to introduce an algorithm for the dicretization of second order elliptic operators in the context of finite volume schemes. The technique consists in matching to a finite volume discretization based on a given mesh, a finite element volume representation on the same given mesh. An inverse operator is also built. This numerical algorithm, which was introduced in [GP1 99], (see also [GP2 99], [GP3 99] and [PAS 99], is based on footbridges between the finite volume representation of functions and a finite element one, both defined on the same mesh. We are going to give some details in the two dimensional case, refering to the aforementioned references for a more complete account and for the general ncf-dimensional case (which is in fact similar to the 2D case). Let £2 be the computational domain that we assume being triangularized : Q = \\ T. Let XT be the piecewise-constant characteristic function defined rer on each triangle T. For each edge A, let us define the piecewise-linear function (f>A such that

Similarly, to each u in £, we first construct v = HVEU in V solution of :

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The finite element representation allow to discretize easily a viscous operator thanks to a variational formulation and the footbridge HVE allow to resume to a finite volume discretization well suited for applying upwinding. But YlyE ° n^v 7^ Idy, then in the case of a zero diffusion parameter, the solution is modified by the use of the two footbridges. We construct another footbridge n from E into V such that II o MEV = Idy. For that purpose, it is sufficient that for each UEF in E to find VVF in V solution of the system :

Our first observations are that HEV is injective and that (89) has no solution in general. Let B be the N F x NE matrix such that Bja = fnXajdx and bEF the NF vector given by bfF = Jn UEF jdx, for j - 1, • • •, NF. The system (89) can then be written in the following way : BuVF = bEF. It is a rectangular linear system with N E unknowns and NF equations. As asymptotically, NS ~ ^ f - , NF ~ 3-/VE72, a least square formulation :

find

minimizing

is used to solve this system. Since the rank of B is NE, (90) has a unique t solution uv F — PVEUEF. Its value, ^BB) BbEF, has low computational t cost since BB is a sparse matrix and a Cholesky factorization can be applied. Hence we get PVE ° ^EV — Idy. 3.9. Non conformal meshes Non conformal mesh techniques were investigated in two situations : • when partitionning the whole domain into several sub-domains, allow a non conformal junction between meshes of adjacent sub-domains (see figure l.b); • use local refinments, keeping the structured type of the main discretization (see figure l.a). Flux schemes have interesting features in these situations :

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Figure 1: Non conformal mesh examples

First, they are based on a finite volume, cell-centered, discretization i.e. no need to interpolate node values). Second, the main part of the algorithm consists in computing numerical fluxes passing through interfaces. So, we have to build once at the beginning an explicit (unstructured cases) or an implicit (structured cases) list of interfaces. At each timestep, the main loop will use this list to compute numerical fluxes and put flux contributions to interface neighbours. For unstructured meshes, the same algorithm can be applied to conformal or non conformal meshes. For structured meshes with local refinments, a treelike structure has been tested with simulated recursive procedures (see figure 2 which represents the tree associated to figure l.a). We refer to Tajchman and Freydier [TAJ 96] for more information. 4. Bibliography [ALO 99]

ALOUGES F., Matrice signe et systemes hyperboliques, in Aspects theonques et pratiques de la simulation numerique de quelques problemes physiques, 249-257, Memoire d'Habilitation, Universite Paris-Sud, 1999.

[ACT 99]

ALOUGES F., GHIDAGLIA J.M. AND TAJCHMAN M., On the interaction of upwinding and forcing for nonlinear hyperbolic

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Figure 2: Tree representation of a local refinment

systems of conservation laws, Prepublication du CMLA and article to appear, 1999. [BOU 98]

[BOU 99]

BOUCKER M., Modehsation numenque multidimensionnelle d'ecoulements diphasiques liquide-gaz en regim.es transitoires et permanents : methodes et applications, These, ENSCachan, France, 1998.

BOUCKER M. AND GHIDAGLIA J.M., On the discretization of non conservative terms in the context of multidimensional flux schemes, in preparation.

[DRE 94]

DREW D.A. AND WALLIS G.B., Fundamentals of Two-Phase Flow Modeling, in Multiphase Science and Technology, Volume 8, Hewitt al Editors, Begell House, Inc., New-York, 1994.

[GAL 96]

GALLOUET T., Rough schemes for complex hyperbolic systems Proceedings of the first symposium on finite volumes for complex applications 15-18 July 1996, 1996, Editions Hermes Paris.

[GHI 98]

GHIDAGLIA J.M., Flux schemes for solving nonlinear systems of conservation laws, Proceedings of the meeting in honor of P.L. Roe, Chattot J.J. and Hafez M. Eds, Arcachon, July 1998, to appear.

Invited speakers [GHI 99]

[GKL 96]

93

GHIDAGLIA J.M., On the numerical treatment of boundary conditions in the context of flux schemes, In preparation.

GHIDAGLIA J.M., KUMBARO A. ET LE COQ G., Une methode volumes-finis a flux caracteristiques pour la resolution numerique des systemes hyperboliques de lois de conservation, C.R.Acad. Sc. Pans, 1996, 322, I, 981-988.

[GKT 97]

. GHIDAGLIA J.M., KUMBARO A., LE COQ AND TAJCHMAN M., A finite volume implicit method based on characteristic flux for solving hyperbolic systems of conservation laws, Proceedings of the Conference on : Nonlinear evolution equations and in finite-dimensional dynamical systems (ShangaT, June 1995), Li Ta-Tsien Ed., 1997, World Scientific, Singapore.

[GLT 99]

[GP1 99]

Ghidaglia J.M., Le Coq G. and Toumi I., Two flux-schemes for computing two-phases flows throught multi-dimensional finite volume methods, NUclear REactor Thermal Hydraulics (NURETH 9), October 1999

GHIDAGLIA J.M. AND PASCAL F., Passerelles volumes finis - elements finis, C.R.Acad. Sc. Pans , I, 328, 711-716, 1999.

[GP2 99]

GHIDAGLIA J.M. AND PASCAL F., Passerelles volumes finis - elements finis, methodes et applications, Rapport EDF/DER/TTA/HT-33/99/002/A, Chatou, France, 1999.

[GP3 99]

GHIDAGLIA J.M. AND PASCAL F., Footbridges between finite volumes and finite elements, Article to appear.

[HAL 98]

HALAOUA K., Quelques solveurs pour les operateurs de convection et leurs applications a la mecamque des fluides diphasiques, These, ENS-Cachan, France, 1998.

[ISH 75]

ISHII M., Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris, 1975.

[KUM 97]

KUMBARO A., Methode VFFC, application au probleme du robmet de Ransom, Rapport EDF/DER/TTA/HT33/97/021/A, Chatou, France, 1999.

[MIM 99]

MlMOUNl S., Quelques cas tests numeriques pour la simulation des ecoulements diphasiques, Rapport EDF/DER/TTA/HT-33/98/031/B, Chatou, France, 1999.

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[PAS 99]

PASCAL F., Passerelles Volumes finis - elements finis, Proceed-

ings of the H eme seminaire sur les ecoulements compressibles et la mecanique des flmdes numerique, CEA Saclay, Janvier 1999. [RAN 89]

RANSOM V.H., Numerical Modeling of Two-Phase Flows, Cours de 1'Ecole d'ete d'Analyse Numerique - CEA-INRIAEDF, 12-23 juin 1989.

[SFW 97]

STADKE H., FRANCHELLO G. AND WORTH B., Numerical simulation of multi-dimensional two-phase flow based on flux vector splitting, Nucl. Eng. Design, 1997, 177, 199-213.

[STE 84]

STEWART H.B., WENDROFF B., Two-phase flows : models and methods, J. Comput. Phys., 1984, 56, 363-409.

[TAJ 96]

TAJCHMAN M. AND FREYDIER P., Methode Volumes Finis a Flux Caracteristiques. Application a un calcul bidimensionnel sur un maillage non conforme, Note, Departement Transferts Thermiques et Aerodynamique, Direction des Etudes et Recherches, Electricite de France, EDF HT-30/96/004/A, May 1996.

[TAJ 98]

TAJCHMAN M. AND FREYDIER P., Schema VFFC : application a I 'etude d'un cas test d'ebullition en tuyau droit representant le fonctionneraent en bomllotte d'un coeur REP. Note EDF HT-33/98/033/A, decembre 1998.

[TOU 92]

TOUMI I., A weak formulation of Roe's approximate solver, J. Comput. Phys., 1992, 102, 360-373.

[TKP 99]

TOUMI I., KUMBARO A. AND PAILLERE H., Approximate Riemann solvers and flux vector splitting schemes for twophase flow Note presented at the 30t/l VKI CFD Lecture Series, 8-12 March 1999.

The MoT-ICE: a new high-resolution wave-propagation algorithm based on Fey's Method of Transport

Sebastian Noelle Institut fur Angewandte Mathematik Universitdt Bonn Wegelerstr. 10 53115 Bonn, Germany [email protected] uni-bonn. de

ABSTRACT The numerical solution of multi-dimensional systems of conservation laws is dominated by up wind-schemes based on one-dimensional Riemannsolvers. Such schemes neglect the physical directions of wave-propagation and replace them by the grid-directions. One of the algorithmic alternatives is Fey's Method of Transport (MoT). Similarly to its one-dimensional fore-runner, the Steger- Warming-Scheme, the MoT suffers from an inconsistency at sonic points. Here we derive a new version of Fey's multi-dimensional flux-vector-splitting and a globally consistent second-order-accurate characteristic scheme based on Interface-Centered-Evolution, the MoT-ICE. Numerical experiments show the stability, accuracy and efficiency of the new scheme. Key Words: Systems of Conservation Laws, Multi-dimensional Flux-VectorSplitting, Fey's Method of Tranport.

1. Introduction Since the work of Godunov, Van Leer, Harten-Lax and Roe, the numerical solution of systems of hyperbolic conservation laws is dominated by Riemannsolver based schemes (see [HIR 90] for the classical references). These onedimensional schemes are extended to several space-dimensions either by using dimensional-splitting on cartesian grids or by the finite-volume approach on un-

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structured grids. For both approaches, convergence and error estimates have been established for multi-dimensional scalar conservation laws, see for example the results and references in [CCL 94, KNR 95, NOE 96, WN 99]. Naturally, there are no comparable results for multi-dimensional systems, since no existence and uniqueness of the p.d.e.'s is known in this case. The first systematic criticism of using one-dimensional Riemann-solvers for multi-dimensional gas-dynamics goes back to Roe himself [ROE 86]: the Riemann-solver is applied in the grid- rather than the flow-direction, which may lead to a misinterpretation of the local wave-structure of the solution. A description of a number of failings of exact and approximate Riemann-solvers for the two-dimensional Euler-equations of gasdynamics may be found in Quirk's paper [QUI 94]. LeVeque and Walder [LW 92] present difficulties of Godunov's scheme for strong two-dimensional Shockwaves arising in astrophysical flows and propose the use of rotated Riemann-solvers. In [ROE 91, NOE 94] Roe and Noelle study oscillations generated by dimensional-splitting-schemes for a prototype linear system. Since the mid-eighties, Roe, Deconinck, Van Leer and many others developed the so-called fluctuation-splitting schemes for the equations of gasdynamics. In these schemes the divergence of the multi-dimensional flux-vector (called fluctuation) is split according to a grid-independent wave model (see [VNL 92] for references). In [NOE 94], the author developed a fluctuationsplitting-scheme for a 2x2 model problem which cures the directional effects exhibited by dimensional-splitting-schemes. However, this fluctuation-splittingscheme seems to be very dissipative for non-stationary problems. Another multi-dimensional approach is the Corner-Transport-Upwind (CTU) scheme of Colella [COL 90]. Here waves are not only propagated to the neighbors which join a common side with a cell, but also to the corner cells. LeVeque's wave-propagation-algorithm, which forms the backbone of the publicly available software-package CLAWPACK, splits one-dimensional fluxes in the transverse directions and propagates them to the corner cells (see [LEV 97] and the references therein). In [BT 97], Billet and Toro introduce weightedaverage-flux (WAF) schemes with good multi-dimensional upwinding- and stability-properties. In this paper we follow the flux-vector-splitting approach, which was first proposed by Steger and Warming [SW 81] in one space-dimension. Instead of decomposing the divergence of the flux-vector, as in the fluctuation-splitting schemes of Roe et.al., the conservative variables and the flux-vector themselves are split (see [HIR 90], Ch.20.2). In his dissertation, Fey developed a multidimensional version of the Steger-Warming-scheme, and called it Method of Transport (MoT). This scheme integrates the acoustic waves over the entire

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Mach-cone (see also [LMW 97] for recent related progress). Subsequently Fey, Jeltsch et.al. simplified the MoT and expanded it in various directions (for references, see [FJMM 97, FEY 98a, FEY 98b]). The starting point of this work are the papers [FEY 98b, FJMM 97]. In these papers, the MoT takes the following form: Step 1. A multi-dimensional wave-model leads to a reformulation of the conservation law as a set of nonlinearly coupled advection equations. Step 2. At the beginning of each timestep, the system is linearized and decomposed into a set of scalar advection equations with variable coefficients. Step 3. The solution of each scalar advection equation at the end of the timestep is computed using a characteristic scheme. Step 4. The solution is projected back onto the conservative variables using the wave-model of Step 1. Our own contribution to Fey's method may be summarized as follows: Given a multi-dimensional wave-model, we simplify Step 2 by distinguishing clearly between linearization and decomposition error. This makes it possible to write down a general second-order-correction term for the decomposition error in a single line. Then we discuss the first-order version of Fey's characteristic scheme (Step 3) and show an inconsistency of the numerical scheme on the scalar level. Subsequently, we derive new first- and second-order characteristic schemes which are globally consistent. Due to the simpler second-ordercorrection term for the decomposition error and a particularly simple choice of numerical transport-velocities our second-order scheme needs only 2.2 times the CPU-time of the first order scheme, and is very efficient. Fey's transport algorithm might be called MoT-CCE, since his scalar scheme uses Cell-Centered-Evolution. We call our new scheme, which is based on Interface-Centered-Evolution, MoT-ICE. In Section 2, we discuss the multi-dimensional wave-models and the linearization and decomposition steps. In Section 3, we show the inconsistency of the MoT-CCE and present the MoT-ICE. In Section 4, we give numerical experiments. Finally, in Section 5, we discuss topics for future research. This is a report on work in progress. The proofs of the theoretical results in Sections 2 and 3 as well as several important implementation details in Section 4 will be given in the forthcoming paper [NOE 99]. The author would like to thank Dr. Michael Fey and Christian von Torne for stimulating discussions on the Method of Transport. This work was supported by DFG-SPP ANumE. The hospitality of Dr. Fey, Prof. Jeltsch and his group during a visit at ETH Zurich is gratefully acknowledged. 2. Decomposition of multi-dimensional hyperbolic systems into scalar advection equations In this section, we recall Fey's advection form and give a general secondorder-accurate linearisation and decomposition of systems which can be written in advection form.

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2.1. A general framework for multi-dimensional flux-vector-splitting Consider a multi-dimensional system of n conservation laws in d space dimensions,

where U : Rd x R+ ->• Rn is the state-vector and f : Rn -> R n x d the fluxvector. The following framework for multi-dimensional flux-vector-splitting (FVS) has been developed by Fey, Jeltsch and collaborators [FJMM 97]: Definition 1 A wave-model for (1) is a set of L > 1 mappings

which satisfy the consistency conditions

and

Given a wave-model, the conservation law (1) may be rewritten in the following advection form:

Wave-models for the wave-equation, the equations of isentropic and non-isentropic gas dynamics and the equations of ideal magneto-hydrodynamics may be found in [FJMM 97]. Common to all of them is a finite set of acoustic waves which approximate the Mach-cone (typically four waves in two spacedimensions). For the Euler-equations one adds an entropy-wave, and for the MHD-equations, the two Alfven and possibly some slow magneto-acoustic waves. 2.2. Decomposition into scalar advection equations Let us see how the advection form can be used when advancing the solution for a single timestep k — At. Ideally, one would like to set each summand in (4) to zero seperately. As we shall see below, this will lead to an error of O(k2) during each time-step, limiting the overall accuracy of the scheme to first-order. The following linearisation will be helpful in order to obtain a second-orderaccurate decomposition:

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Lemma 2 (linearisation) Let U be a smooth solution of (1), k = At > 0. Let

satisfy

and

Then

The proofs of this and the following results will appear in [NOE 99]. Note that the approximate transport velocities a/ are now prescribed coefficients which depend on space but not on time. System (5) is still nonlinear in V. It is, however, linear in the components Sj(V), since V = 53j=1 Si(V). Next let us decompose the linearized advection form (5) by setting each summand to zero seperately: Let

satisfy

Then

where

is given by

Therefore, this naive ansatz leads to a first-order decomposition error. Taking a closer look at the leading part of the error, we note that

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since

This can be used in order to derive a second-order-accurate decomposition of (5) respectively (4). Theorem 3 (second-order-accurate decomposition) Let

satisfy

Then For the Euler equations and the shallow-water equations, Morel, Fey and Maurer have also derived a second-order-accurate decomposition into linear advection equations. Instead of evaluating the velocities at the half-timestep, as done in Lemma 2, they freeze them at the original timestep. As a consequence, the linearization- and decomposition-errors are not seperated, and the secondorder-correction terms become more involved. 3. Solving the scalar advection equations by characteristic schemes The linearisation and decomposition given in the previous section leads us, at the beginning of each timestep, to a set of scalar transport equations of the form

For the rest of this section, we consider the velocity-vector a : Rd x R+ —>• Rd to be a given function of x and t. The function (p : Rd x R+ —>• R is the unknown, and initial data are prescribed at t = tn := nk,

These equations may be solved by introducing characteristics z(r;x,t) by

z : R+ x Rd x R+ ->• Rd z(t;x, £) = x 0 T z(r;x,t) = a(z,r). Since the flux-vector

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Figure 1: Backwards characteristic transport of cell Kij. The region bounded by the curved lines is z(tn;Kij,tn+i), and the dashed lines are backwards characteristic curves issuing from the corners of cell K^ at time tn+\.

Lemma 4 For all K C Rd, t, r € R+

3.1 Characteristic schemes We would like to use Lemma 4 to construct numerical methods for solving (6). For simplicity, we restrict the analysis from now on to two spacedimensions, and use the notation x — (x, y) G R2 for the space variable and a = (a, b} e R2 for the velocity field. Let KIJ C R2 be the cells of a uniform cartesian grid with mesh-size h — Ax = Ay, and recall that A; = At is the timestep. Due to Lemma 4,

Note that Ki>j> fl z(tn;Kij,tn+i) is that part of KVJI that will be mapped to Kij by the characteristic flow from time tn to time tn+i (compare Figure 1). Definition 5 A characteristic scheme for (6) is given by

where (f>n is a piecewise smooth spatial reconstruction of <£>(•, tn), and K*L, an approximation of KVji n z(tn; Kij, tn+i).

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For the piecewise smooth spatial reconstruction of (p(-,tn) we will either use piecewise constant or piecewise linear functions. We will label the corresponding schemes P0 resp. PI. For the moment, let us consider schemes using piecewise constant reconstructions and focus on the approximation of the characteristic flow. 3.2 Cell-Centered-Evolution: the MoT-CCE In a series of papers (see [FEY 98b, FJMM 97] and the references therein), Fey and collaborators have used the following approximation of the characteristic flow, which we will call Cell-Centered-Evolution: In each cell Ki>ji, consider the local characteristic flow defined by

Then set In one space-dimension this leads to the following algorithm:

where A := k/h and the subscripts ± denote the positive resp. negative part of a quantity. Fey calls his scheme MoT ("Method of Transport"), and therefore we will use the acronym MoT-CCE-PO. Let us discuss the consistency of the MoT-CCE-PO with the differential equation (6). If a is constant, then (7) coincides with the first-order upwind scheme. For variable coefficients the situation is more complex: Example 6 Let us consider the case

corresponding to a compressive wave

For constant initial data,

the exact solution remains constant in space but grows exponentially in time,

The approximate solution produced by the MoT-CCE-PO is

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As k tends to zero with X fixed, this solution converges to

Thus (7) is inconsistent with the differential equation (6) at the "sonic" point x — 0, where the transport velocity a changes sign. The cell-centered-evolution leads to an inconsistent approximation of V • a, which takes the value 2 instead of 1. An analogous inconsistency occurs for the case of an expansive wave, V • a > 0. We omit the details. This difficulty was already described by Steger and Warming [SW 81] in 1981: the numerical flux produced by their splitting is not continuously differentiable at sonic points for the equations of gas-dynamics. This results in so-called "glitches" at sonic points. Subsequently, Van Leer developed a splitting with continuously differentiable fluxes (compare [HIR 90], Ch. 20.2.3). In her dissertation, Morel [MOR 97] also observed glitches at sonic points for two-dimensional shallow-water-computations carried out with the MoT-CCEPO, and generalized the Van-Leer-flux-vector-splitting to two space-dimensions in order to remove these numerical artefacts. However, this slows down the algorithm considerably, and Morel herself remarks that her method does not seem to be generalizable to second-order-accuracy. Finally, let us remark that in Example 6 above, the MoT-CCE-PO diverges in the L°°-norm, but converges in I/1. Moreover, this inconsistency does not occur when one approximates the velocity field a by piece wise linear functions. 3.3 Interface-Centered-Evolution:

the MoT-ICE

In the following we will derive a consistent alternative to the cell-centeredevolution discussed above. Since the new scheme will approximate the transport velocities at the interfaces between the cells, we will call it MoT-ICE for Interface-Centered-Evolution. Instead of using a cell-centered approximation of the local characteristic flow to decompose Kij, we now define auxiliary transport velocities an 1 . on + 2 '3 the vertical interfaces l

resp. bn.. ! on the horizontal interfaces J

J-T

2

between the cells. Let A = k/h be fixed and suppose that the CFL-condition

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Figure 2: Auxiliary transport velocities and approximate characteristic decomposition of a two-dimensional cell KIJ via Interface-Centered-Evolution.

is satisfied. In order to approximate the backwards characteristic flow for one timestep, we shift the vertical interfaces ^ j+i horizontally by (—k a" i .) and the horizontal interfaces JIJ+L vertically by (—k bn.

L ).

Depending on the

signs of the velocities a and 6, each cell KIJ will be divided into one up to nine sub-rectangles, leading to a natural subdivision

(compare Figure 2). For example,

and so on.

3.3.1. The MoT-ICE-PO In order to define the MoT-ICE-PO, we approximate (f> by piece wise con stants and define the auxiliary transport velocities by

where the remainders ai+i^ and fc^j+i are supposed to be lipschitz-continuous. Theorem 7 Suppose that \ — k/h is fixed, and that the CFL-condition (8) is satisfied. Then for any given smooth velocity field a : R2 x R+ —> R2, the MoT-ICE-PO is consistent of order one with the differential equation (6).

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Note that the theorem states global first order consistency. The difficulty is to control the truncation error near points where the characteristic velocities change sign. See [NOE 99] for the full proof. 3.3.2. The MoT-ICE-Pl Let us next construct a scheme based on piecewise linear reconstructions, the MoT-ICE-Pl. We have to define the auxiliary transport velocities at the interfaces and the piecewise linear reconstruction of the solution (p. We denote the piecewise linear reconstruction of R\ For ( x , y ) G Kij let

A rather involved computation leads to the following choice of auxiliary transport velocities at the interfaces which guarantees second order consistency for smooth solutions:

where a and b are supposed to be lipschitz-continuous. Let us give a geometrical interpretation of these terms: the factors (a 4- fat) resp. (b + |&t) simply reflect the fact that we wish to evaluate the transport velocities a resp. b in the center of the space-time interfaces II+LJ x [£n,*n+i] resp. Jij+L x [£ n ,t n+1 ] in order to achieve second-order accuracy in time. The factor (1 — |(az -I- 6 y )) which appears both in a and 6, is an approximation of

at t = k/1. This approximates the evolution of (p along the characteristics z(r;x,i), where (6) gives

The correction terms f (&&) y resp. f (ab)x account for fluxes across corners of the grid, see [NOE 99] for a more detailed discussion. Let us now state the consistency of the MoT-ICE-Pl: Theorem 8 Let a : R2 x R+ —> R2 be a given smooth velocity field and let (p : R2 x R+ —>• R be a smooth solution of (6). Suppose that there are lipschitz-continuous functions ^p"x and ^ such that the piecewise linear reconstruction (pn defined in (9) satisfies

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Let the auxiliary transport velocities ai+i j and bij+i be given by (10) (11), let A = k/h be fixed and suppose that the CFL-condition (8) holds. Then the MoT-ICE-Pl is consistent of order two with the differential equation (6). Once more, we refer to [NOE 99] for the proof, which requires a careful study of the truncation errors of the piecewise linear reconstruction (pp_. Let us stress again that we do obtain second-order consistency at all points, including those where the transport-velocities change sign.

4. Numerical experiments

In this section we present numerical experiments which confirm the accuracy and stability of the new MoT-ICE for smooth and discontinuous solutions. For the piece wise-linear reconstruction, we choose a central version of the WENO (Weighted Essentially Non-Oscillatory) reconstruction [JS 96]. Details may be found in [NOE 99]. 4.1 Scalar advection with periodic

coefficients

In order to illustrate the failure of consistency of the MoT-CCE-PO (cellcentered evolution) and the consistency of the new MoT-ICE (interface-centered evolution), we consider the one-dimensional scalar advection equation

over the interval [—1,1] with

and

Note that at x = 0, v?(0,£) satisfies the ordinary differential equation

so

Similarly, and these are the maximum and minimum of the exact solution. We compute the solution at time T — loe(2)/7r, so

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Table 1: EOCs for one-dimensional advection with periodic coefficients

ix 40 80 160 320 640 ix 40 80 160 320 640 ix 40 80 160 320 640

MoT-CCE-PO EOC LOQ Lj. 3.439066e-01 5.456309e-02 3.210264e-02 0.77 4.215299e-01 1.731594e-02 0.89 4.617376e-01 8.996724e-03 0.94 4.813319e-01 4.587514e-03 0.97 4.908176e-01 MoT-ICE-PO EOC Li LOO 8.834546e-02 3.075886e-02 1.599486e-02 0.94 4.741382e-02 8.114853e-03 0.98 2.471667e-02 4.094363e-03 0.99 1.263470e-02 2.055908e-03 0.99 6.379015e-03 MoT-ICE-Pl with WENO-limiter EOC Li ^00 5.123679e-03 2.193228e-03 2.04 1.532804e-03 5.330401e-04 1.281400e-04 2.06 3.878244e-04 2.993688e-05 2.10 7.604266e-05 5.934229e-06 2.33 1.523297e-05

EOC -0.29 -0.13 -0.06 -0.03 EOC 0.90 0.94 0.97 0.99 EOC 1.74 1.98 2.35 2.32

We choose a CFL-number of 41og(2)/yr, roughly 0.88. In Table 1, we list the experimental orders of convergence (EOCs) both with respect to the Ll and L°° norm. In the L^ncrm, the EOC of the MoT-CCE-PO starts at 0.77 and increases towards 1. However, the method diverges in L°°, as it should have been expected from Example 6 in Section 3.2. Contrary to that, the MoT-ICE converges uniformly (i.e. in L1 and I/00) to the expected orders. Comparing the L1 errors of the MoT-CCE-PO and the MoT-ICE-PO (piecewise constant reconstructions) , one sees that the MoT-ICE converges with a better rate (especially on the coarser grids) and produces roughly half the error of the MoT-CCE. The convergence rates of the MoT-ICE-Pl (piecewise linear reconstructions) are even better than 2 on the finer grids, both for the unlimited and the limited version (we omit the table for the unlimited scheme). The error of the scheme using WENO-limiter is only slightly larger than that of the unlimited scheme and it is of course orders of magnitude smaller than that of the first order MoT-ICE-PO.

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4.2 Rotating Smooth Hump Our next test problem is the two-dimensional scalar equation (6) with

in the domain [—1,1]2. Note that on this problem, the MoT-CCE-PO and the MoT-ICE-PO will produce identical results, since ax = by = 0, so

and

We compute one full rotation, i.e. T = TT, and fix A = At/Ax = Tr/6. Assuming that the maximal transport velocity is \/2, this corresponds to a CFL-number of roughly 0.74. First, we consider smooth initial data. Let XQ := 0.5, yo := 0, TO := 0.3,

Table 2: EOCs for smooth rotating hump. ix 40 80 160 320 640 ix 40 80 160 320 640 ix 40 80 160 320 640

MoT-ICE-PO EOC EOC Height Li LOO 9.528519e-01 0.287 7.129756e-01 0.454 6.479957e-01 0.56 5.4592186-01 0.39 3.994583e-01 0.70 3.690224e-01 0.631 0.56 2.277596e-01 0.81 2.228099e-01 0.73 0.777 1.228174e-01 0.89 1.239005e-01 0.85 0.876 MoT-ICE-Pl with unlimited central differences EOC EOC Height Li Loo 0.794 2.596250e-01 2.0565206-01' 2.24 4.491245e-02 2.20 0.961 5.505040e-02 1.091899e-02 2.33 8.892373e-03 2.34 0.995 2.426323e-03 2.17 1.967944e-03 2.18 0.999 5.822946e-04 2.06 4.673955e-04 2.07 1.000 MoT-ICE-Pl with WENO-limiter EOC EOC Height Li Loo 4.351708e-01 0.525 4.749751e-01 0.767 1.201747e-01 1.86 2.327933e-01 1.03 3.337591e-02 1.85 1.018161e-01 1.19 0.898 0.960 6.350393e-03 2.39 4.022132e-02 1.34 0.988 1.030164e-03 2.62 1.230235e-02 1.71

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and otherwise.

For the MoT-ICE-PO the EOCs increase slowly towards unity both in L1 and L°° (see Table 2). The error is very large on the coarser grids, and the convergence is initially slow. This is natural, since the initial data are not well resolved on the coarse grids, where they appear rather as a sharp peak than a smooth hump. For the MoT-ICE-Pl with unlimited central differences the EOCs are better than two both in L1 and L°° and they converge towards two as the grids are refined. For the MoT-ICE-Pl with WENO-Limiter, the EOCs in L1 are slightly below two in the beginning. However, as the grid is refined, the EOCs increase drastically and well beyond two. In L°°, the EOCs start slightly above unity on the underresolved coarse grids, but show a similar dramatic increase as the grids are refined. In our experience, this behavior is typical for the central WENO-limiter.

4.3 Rotating Cylinder Next, we consider the rotating cylinder:

othrewise

Since the solution is discontinuous, we only give the experimental orders of convergence in L1 (see Table 3). We also display the maximal height of the cylinder, in order to see if it is excessively smeared, or whether there are overshoots in the numerical solution. For the MoT-ICE-PO, the EOCs tend towards 0.5 as expected for a linear problem and a scheme based on piecewise constant reconstructions. The maximal height of the cylinder increases towards 1.0 as the grid is refined. For the unlimited MoT-ICE-Pl (we omit the table), the EOCs are decreasing from 0.78 towards 0.70. As should be expected for a

Table 3: EOCs for rotating cylinder. ix 40 80 160 320 640

MoT-ICE-PO EOC Height Li 7.012658e-01 0.868 5.046964e-01 0.47 0.968 3.618134e-01 0.48 0.990 2.583896e-01 0.49 0.995 0.997 1.837913e-01 0.49

MoT-ICE-Pl, WENO-limiter EOC Height Li 3.758964e-01 0.993 1.000 2.278516e-01 0.72 0.70 1.000 1.398394e-01 8.588859e-02 1.000 0.70 5.270885e-02 0.70 1.000

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Figure3: Rotating cylinder. MoT-ICE-PO and MoT-ICE-Pl, 160 x 160 points.

discontinuous solution computed with unlimited piecewise-linear reconstructions, there is an overshoot of 9 to 11 percent of the height of the cylinder. The MoT-ICE-Pl with WENO-limiter converges at rate 0.70, produces no overshoots and only slightly larger L1-errors than the unlimited scheme. As can be seen from the maximal height especially on the coarser grids, the computation is much less smeared than the one with the MoT-ICE-PO. The error on the finest grid is a factor 3.5 smaller for the limited MoT-ICE-Pl than for the MoT-ICE-PO. Figure 3 shows that the cylindrical shape of the solution is nicely preserved by both versions of the scheme. 4.4 Shallow-Water-Equations In her dissertation [MOR 97], Morel reports glitches at sonic points for a radially symmetric explosion for the shallow-water equations computed with the MoT-CCE-PO. Results using the new MoT-ICE, both PO and PI, show no such glitches, and produce almost perfectly radially symmetric solutions (see Figure 4). From these pictures, the results of the MoT-ICE-Pl seem to

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Figure 4: Explosion problem for the shallow water-equations. MoT-CCE-PO, MoT-ICE-PO, MoT-ICE-Pl. Grid of 160 x 160 Points. Left Column: 25 Contours of water-height. Right Column: cross-sections along the z-axis "+" and the diagonal "x".

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be of the same quality as those computed by Morel using CLAWPACK (see [NOE 99] for crucial implementation-details for the MoT-ICE-Pl for systems of conservation laws).

4.5 Comparison of

efficiencies

Let us give a first comparison of efficiencies. Morel reports that the MoTCCE-PO (which is inconsistent at sonic points), Van Leer's flux-vector-splitting and CLAWPACK T1'0 (using the first-order Roe-solver without transverse wave-propagation) all use the same amount of cpu-time (say one time unit). CLAWPACK T1'1 takes 1.5 units, and the first-order fix at sonic points proposed by Morel 2.9 units. Our preliminary experience with the MoT-ICE is the following: the MoT-ICE-PO, which is consistent at sonic points, takes 0.9 units and is hence as fast as standard first order schemes. The MoT-ICE-Pl takes 2.2 units, which is the sames as the second order CLAWPACK T 2 ' 2 . This compares favorably with the MoT-CCE-Pl, which is consistent at sonic points, but needs 10.5 units of cpu-time. 5. Discussion The new Method of Transport with Interface-Centered-Evolution (MoTICE) is consistent at sonic points, second-order-accurate for smooth solutions and nonoscillatory at discontinuities. In all test-calculations carried out by the author so far it produces high-resolution approximations of multi-dimensional phenomena with almost no grid-orientation effects. It needs about the same CPU-time as a second order version of LeVeque's wave-propagation algorithm CLAWPACK which, according to numbers given by Morel, is 4 to 5 times faster than the original second-order-accurate MoT-CCE-Pl. In ongoing joint work with von Torne, we plan to develop the MoT-ICE into a fully adaptive code, capable of handling general geometries. The multidimensional core of the Method of Transport are Fey's wavemodels. Both for theoretical and practical reasons, it would be desirable to establish a systematic approach for their derivation. Are there wave-models for any hyperbolic system of conservation laws, or do they require special symmetries, which are satisfied by the fluid-dynamical equations for which wavemodels have been found so far? Which properties should a good wave-model have, besides the formal consistency-requirements (2)-(3)? Can one analyze the interplay between properties of the wave-model and the stability and efficiency of the resulting wave-propagation algorithm? Let me make one final remark: I would very much like to see a more widespread continuation of the "Great Riemann-Solver Debate" started by Quirk [QUI 94]. Are the failures of Riemann-solver-based schemes reported by Quirk of a cosmetic nature, visible only in the "picture-norm", or can they

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be quantified? Do the schemes diverge or converge when the grid is refined? How do the so-called "genuinely multi-dimensional" schemes perform in these situations? 6. Bibliography For lack of space, we do not attempt to give a well-balanced (much less a comprehensive) bibliography. However, the following list should contain lots of useful information: [BT 97]

BILLET, S. AND TORO, E., On WAF-type schemes for multidimensional hyperbolic conservation laws. J. Comput. Phys. 130 (1997), 1 -24.

[CCL 94] COCKBURN, B., COQUEL, F. AND LfiFLOCH, P., Convergence of the finite volume method for multidimensional conservation laws. SIAM J. Numer. Anal. 32 (1995), 687 - 705. [COL 90] COLELLA, P., Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 87 (1990), 171 - 200. [FEY 98a] FEY, M. Multidimensional upwinding. I. The method of transport for solving the Euler equations. J. Comput. Phys. 143 (1998), 159 - 180. [FEY 98b] FEY, M. Multidimensional upwinding. II. Decomposition of the Euler equations into advection equations. J. Comput. Phys. 143 (1998), 181 - 199. [FJMM 97] FEY, M., JELTSCH, R., MAURER, J. AND MOREL, A.-T., The method of transport for nonlinear systems of hyperbolic conservation laws in several space dimensions. Research Report No.97-12, Seminar for Applied Mathematics, ETH Zurich (1997). [HIR 90]

HIRSCH, C., Numerical computation of internal and external flows. Vol.2, Wiley-Interscience 1990.

[JS 96]

JIANG, G.-S. AND SHU, C.-W., Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126 (1996), 202 - 228.

[KNR 95] KRONER, D., NOELLE, S. AND ROKYTA, M., Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. Numer. Math. 71 (1995), 527-560. [LEV 97] LEVEQUE, R.J., Wave Propagation Algorithms for Multidimensional Hyperbolic Systems. J. Comput. Phys. 131 (1997), 327 - 353.

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[LW 92]

LEVEQUE, R.J. AND WALDER, R., Grid alignment effects and rotated methods for computing complex flows in astrophysics, eds. J.B. Vos, A. Rizzi and I.L. Rhyming, Proceedings of the Ninth GAMM Conference on numerical methods in fluid mechanics. Notes Num. Fluid. Mech. 35 (1992), 376-385.

[LMW 97] LUKACOVA-MEDVIDOVA, M., MORTON, K. AND WARNECKE, G. Evolution Galerkin methods for hyperbolic systems in two space dimensions. Report 97-44, Univ. Magdeburg, Germany (1997). [MOR 97] MOREL, A.-T., A genuinely multidimensional high-resolution scheme for the shallow-water equations. Dissertation, ETH Zurich Diss. No. 11959 (1997). [NOE 94] NOELLE, S., Hyperbolic systems of conservation laws, the Weyl equation, and multi-dimensional upwinding. J. Comput. Phys. 115 (1994), 22- 26. [NOE 96] NOELLE, S., A note on entropy inequalities and error estimates for higher order accurate finite volume schemes on irregular families of grids. Math. Comp. 65 (1996), 1155 -1163. [NOE 99] NOELLE, S., The MoT-ICE: a new high-resolution wavepropagation algorithm for multi-dimensional hyperbolic systems of conservation laws based on Fey's Method of Transport. To be submitted for publication, 1999. [QUI 94]

QUIRK, J., A Contribution to the Great Riemann Solver Debate. Int. J. Numer. Meth. Fluid Dyn. 18 (1994), 555 - 574.

[ROE 86] ROE, P., Discrete models for the numerical analysis of timedependent multidimensional gas dynamics. J. Comput. Phys. 63 (1986), 458 - 476. [ROE 91] ROE, P., Discontinuous solutions to hyperbolic systems under operator splitting. Numer. Meth. Part. Diff. Eq. 7 (1991), 207. [SW 81]

STEGER, J. AND WARMING, R., Flux vector splitting of the inviscid gas-dynamic equations with applications to finite difference methods. J. Comput. Phys. 40 (1981), 263 - 293.

[VNL 92] VAN LEER, B., Progress in multi-dimensional upwinding. ICASE Report 92-43 (1992). [WN 99]

WESTDICKENBERG, M. AND NOELLE, S., A new convergence proof for finite volume schemes using the kinetic formulation of conservation laws. Accepted for publication in SIAM J. Numer. Anal. (Feb. 1999).

Numerical Analysis

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Error estimate for a finite volume scheme on a MAC mesh for the Stokes problem Philippe Blanc CMI, Universite de Provence Technopole de Chateau Gombert, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France email: [email protected]

Abstract We consider a finite volume scheme on MAC mesh for the Stokes equations. Under regularity assumptions on the solution, we prove an error estimate of order 2 in the L2 norm for the pressure and Hl norm for the velocity. Keys words

Finite volume scheme, MAC mesh, Stokes equations.

1. Introduction In this paper, we study some error estimate for the velocity in norm H1 and for the pressure in the L2 norm for the Stokes equations, discretized by a finite volume method on the "Marker and Cell" (MAC) mesh. This mesh was introduced by Harlow and Welch [HAR 65] in the middle of the sixties. Since then, the convergence of different schemes constructed on the MAC mesch has been studied for several methods. One of the first results for the convergence of the MAC scheme was found by Porshing [FOR 78] in 78. More recently in 1997, Shin and Strikwerda found an Inf-Sup condition [SHI 97] and an error estimates [SHI 96] for finite difference approximations. Two other methods are used on a MAC mesh: the mixed finite element method and the finite volume method. An example of mixed finite element method is given by Farhloul and Fortin in an article [FAR 97] in 1997. And in 1998, Han and Wu [HAN 98] proved a first order error estimate for both the velocity and the pressure. There exist several finite volume methods, some examples have been given, in 1983, in the book of Peyret and Taylor [PEY 83]. One of these has been studied by Nicolaides [NIC 92] in 1992 : the covolume method. This method

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allows the use of some results proved for "div-curl" systems. And in 1997, Nicolaides and Wu [NIC 97] proved a first order error estimate for the velocity and the pressure. In 1998, Chou and Kwak [CHO 98] introduced a covolume method for the generalized Stokes problem, and proved a first order estimate for the velocity and the pressure, provided that the exact velocity is in H2 and the exact pressure is in Hl. We choose here to directly discretize the velocity and pressure in the Stokes equations by a finite volume method, which we describe in the second part. Finally in the third part, we will prove a second order error estimate for the velocity and the pressure, provided that the exact velocity is in C3 and the exact pressure is in C2. 2. The finite volume scheme The purpose of this work is the study of a finite volume scheme for the discretization of the following Stokes problem:

where v is the viscosity, (w, v) is the velocity, p is the pressure and (/, g) is the given force. And we assume: Assumption 1 (i) (H) (Hi)

17 is an open bounded polygonal convex subset of M2, v > 0, /<EL 2 (17) and g € L2 (17)

We consider a MAC mesh on 17 (cf figure 1) satisfying: Assumption 2 There exists £ > 0 such that for all i and all j, we have:

(i) (ii) (Hi) (iv)

hi > C/i, kj > C/i, /ii+i/2 > (h, kj+i/2 > C^-

Numerical analysis

Figure 1: MAC mesh Then we write the finite volume scheme:

where

and for all

119

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Finite volumes for complex applications

and for all

This scheme [2]-[6] has a solution, which is unique up to an additive constant for the pressure (see e.g. [EGH 97]).

3. Error estimates

3.1. Notations Let (u,v,p) be the solution of the Stokes equation satisfying:

and we assume: Assumption 3 ue C 2 (O), veC2(fy

andpeCl(ti).

Then we denote: Notation 1

Let (uT,vT,pT) be the solution of the scheme [2]-[6] satisfying:

then we denote the error: Notation 2

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121

Remark 1 We have:

Notation 3

For a mesh T, XC'7") denotes the space of functions which are constant on each cell o f T .

3.2. Error estimates We prove here the following error estimate: Theorem 3.1

Under the assumptions 1 and 2, ifu G C 2 (fJ), v G C2(Cl), andp G Cl(Cl}, there exists C depending only of the exact solution (u, v,p) and fi such that:

Futhermore ifu € C 3 (fi), v € C*3((7), p G C 2 (fJ) and if tie mesh is uniform, then we have:

where ||e"||i)Tu and ||e^||i )Tv are the errors for the velocity in the discrete HQ norm (see [EGH 97] or [EGH 99] for the exact definition) and ||e£|| is the error for the pressure in the L2-norm. 3.2.1. Error estimate for the velocity Lemma 3.1

Under the assumption 1 and 3, and with the notations 2, there exists C\ depending only on (u, v,p] and on fi such that, for all a > 0, we have:

Furthermore ifu G C 3 (fJ), v G C3(Cl), p G C2(Cl) and if the mesh is uniform, then we have:

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To prove the first part of this lemma, we replace (ui+-\./2,j^vi,j+i/2jPi,j) m the scheme by (e"+1/2 - ' e i 7+1/2' e ??)> then we introduce some second order terms. Then we multiply each equation respectively by eV, 1,2 •, e" •+i/<2 and e? -, we sum over i and j, and we add them. We obtain, using a discrete inetegration by part:

thanks to Young's inequality and the regularity of u and v, one has:

and for all a G R, a > 0, thanks to Young's inequality and the regularity of p, we have:

Taking back [11] and [12] in [10], we obtain the result. For the second part, we use the regularity of (w,v,p) and of the mesh to replace the second order terms by some third order terms, changing all "h 2 " by "h 4 ". 3.2.2. Error estimate for the pressure Lemma 3.2

Under the assumptions 1, 2 and 3, and with the notation 2, there exists Ci depending only on (u, v,p) and fi such that:

Ifu£C3(Cl),v£

C 3 (fJ), p 6 C 2 (O) and if the mesh is uniform, then:

To prove this lemma, we use the following theorem proved by Shin and Strikwerda [SHI 97]. Theorem 3.2

Under the assumption 2, there exists C/ depending only of ft such that for all q e X(T), there exists (u,v) G X(T U ) x ~X(TV] satisfying:

Numerical analysis

123

Applying this theorem to e?r we obtain a "velocity" (u, v) € X(T U ) x %(T V ). Then we replace ( u i + i / 2 , j ^ i , j + i / 2 j P i , j ) m the first and the second equations of the scheme by (e"+1/,2>J., e^-+1/2, ef^-), we multiply each respectively by u i+ i/2,j and by ^-+1/2, and we sum each equation on i and j and we add both to obtain:

with, thanks to the Cauchy-Schwarz inequality:

where C\ only depends on (u,v,p) and £1. Hence from [16],[17], [18] and [15] and thanks to Theorem 3.2, there exists C > 0 such as:

so by Lemma 3.1, with .

the result. For the second part of the

i.e. with 2

4

lemma, we replace again all "/i " by "/i " in the preceding reasoning. 3.2.3. Proof of theorem 3.1 The error estimate for the pressure is given by Lemma 3.2. For the velocity, we take back the value of a and C^ in the estimate [8], and we obtain:

Therefore the theorem is proved with C = max(C2, 64).

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References [CHO 98] CHOU S.H. AND KWAK D.Y., A Covolume method Based on Rotated Bilinears for the Generalized Stokes Problem. SIAM J. Numer. Anal, Vol. 35, N° 2, p. 494-507, April 1998. [EGH 99] EYMARD R., GALLOUET T. AND HERBIN R., Convergence of finite volume schemes for semilinear convection diffusion equations. Numer. Math., Vol. 82, p. 91-116, 1999. [EGH 97] EYMARD R., GALLOUET T. AND HEREIN R., Finite Volume Methods. Preprint N° 97-19 LATP, Aix-Marseille 1, to appear in Handbook of Numerical Analysis, P.O. Ciarlet, J.L. Lions eds. [FAR 97] FARHLOUL M. AND FORTIN M., A New Mixed Finite Element for the Stokes and Elasticity Problems. SIAM J. Numer. Anal, Vol. 30, N° 4, p. 971-990, August 1997. [HAN 98] HAN H. AND Wu X., A New Mixed Finite Element Formulation and the MAC Method for the Stokes Equations. SIAM J. Numer. Anal, Vol. 35, N°. 2, p. 560-571, April 1998. [HAR 65] HARLOW F.H. AND WELSH J.E., Numerical calculation of timedependant viscous incompressible flow of fluid with free surfaces. Phys. fluids, Vol. 8, p. 2181-2189, 1965. [NIC 92]

NiCOLAlDES R.A., Analysis and Convergence of the MAC Scheme I. The Linear Problem. SIAM J. Numer. Anal, Vol. 29, p. 15791591, 1992.

[NIC 97]

NiCOLAlDES R.A. AND Wu X., Covolume Solutions of ThreeDimensional Div-Curl Equations. SIAM J. Numer. Anal, Vol. 34, N° 6, p. 2195-2203, December 1997.

[PEY 83] PEYRET R. AND TAYLOR T.D., Computational Methods for Fluid Flow. Springer- Verlag, New York, 1983. [POR 78] PORSHING T. A., Error Estimates for MAC-Like Approximations to the Linear Navier-Stokes Equations. Numer. Math., 29, p. 291-306, 1978. [SHI 96]

SHIN D. AND STRIKWERDA J.C., Convergence Estimates for Finite Difference Approximations of the Stokes Equations. J. Aust. Math. Soc. Ser B, 38, p. 274-290, 1996.

[SHI 97]

SHIN D. AND STRIKWERDA J.C., Inf-sup Conditions for Finite Difference Approximations of the Stokes Equations. J. Austral. Math. Soc. Ser. B, Vol. 39, p. 121-134, 1997.

Convergence Rate of the Finite Volume Timeexplicit Upwind Scheme for the Maxwell System on a Bounded domain

Yves Coudiere INSA, complexe scientifique de Rangueil, 31077 Toulouse Cedex 4, France P. Villedieu ONERA Toulouse, 2 Avenue Edouard Belin, 31055 Toulouse Cedex, France

ABSTRACT : We derive an O(h1/2) error estimate for the upwind, explicit in time, finite volume scheme for Friedrichs systems. Explicit schemes in that case can not be seen as standart time-space finite element ones. Our demonstration is the generalisation on bounded domains of the ideas of Vila and Villedieu for the Cauchy problem in Md. It is applied to the case of Maxwell's equations. Key Words: hyperbolic system, Maxwell's equations, finite volumes, error estimates.

1. Introduction We are interested in the approximation by finite volume means of the Friedrichs systems of the form

d

where Ai are some symmetric matrices,An = 2_.^-ini (n being the unit nor»=i mal outward to £)). (1) belongs to the class of hyperbolic systems first introduced by Friedrichs [FRI 58]. For M such that ker (An — M) is maximal positive, it have a unique solution in L2(]0, T[x£}) [RAU 85].

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Finite volumes for complex applications

We shall treat, as an example, the case of the two dimensional Maxwell's equations in Transverse Magnetic Mode. Our aim is to derive an O(/i 1//2 ) error estimate for the upwind, explicit in time, finite volume scheme for (1), under very general assumptions on the mesh and minimal regularity hypothesis on the continuous solution u. The discontinuous Galerkin method, applied to the approximation of solutions of (1) is a space-time finite element method, which can be interpreted as a finite volume method, but with an implicit in time discretisation [JOH 87]. However, finite volume schemes with time-explicit discretisation can not be interpreted in general as discontinuous space-time finite element schemes associated to a coercive bilinear form. Derivation of error estimates of order /i 1 / 2 for such schemes in the case of the Cauchy problem on Md was first obtained by Vila and Villedieu [VIL 97, VIL 99], using a new technique of demonstration. On bounded domains, the additional difficulty is the discretisation of the flux Ann on the boundary of Q. Here, we propose a general form for the numerical flux on d£l, which guarantee the consistency and the stability of the corresponding scheme. We also prove an error estimate of order /z 1 / 2 (as for time implicit schemes). In section 2, we present the Maxwell's equations under the form (1). The scheme is defined in section 3. The actual results are stated in section 4. Sketches of the proofs can be found in section 5.

2. The Maxwell system As an example, we consider the bidimensional Maxwell system in Transverse Magnetic mode (TM waves). It may be written as a Friedrichs system of the form (1) in Q C M2, with

The unknown is u = [ E Hx Hy ] (Electric an Magnetic Fields; supposing that c = 1 is the light speed). We shall suppose that fi is the bounded domain between an obstacle and an outer boundary. Classical boundary conditions are - on the obstacle (d£l\), the metallic condition: E = 0,

Numerical analysis

127

- on the outer boundary (dO,2), the linearized absorbing boundary condition of Silver Muller: H A next = E. These conditions may be stated like in (1) by taking

where the outward normal to <9Q is n ex t = [ ® P ] •

3. The numerical scheme

Let Th be a mesh of Q, composed of polyhedral cells K. In order to avoid any local degeneracy of the mesh, we assume that there exists some positive constants a and b such that

m(K), m(dK), diam(A') denote, respectively, the measure in Md of the cell K, the measure in M d - 1 of the boundary of K, the diameter of K. Let At be a time step, and tn = nAt. We shall approximate the solution of (1) by a piecewise constant function Vh such that

The values of i>/j are calculated according to the following scheme:

g^e are some numerical fluxes, defined below; B+ and B denote, respectively, the positive and the negative parts of a symmetric matrix B.

3.1 The interior numerical flux Let «5£ be the set of the interfaces interior to Q. We take the natural upwind scheme on such interfaces:

3.2 The numerical flux on the boundary

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Finite volumes for complex applications

Let dSh be the set of the edges of the boundary 5Q. We propose to take the following general form for the numerical flux on the boundary:

Since An = - (An + M) + - (An — M) and (An - M) u = 0 on the boundary, Zi Zt a natural choice is -yeQe = 0- But it yields an non stable scheme in general. • Qe is a stabilization term that does not modify the consistency of the flux. We shall take: Qe is the perpendicular projection on ker (Ane — Me) . • 7e > 0 is a parameter measuring the importance of the stabilization term. It shall have an effect on the CFL condition.

3.3 Case of the Maxwell system For the maxwell system given by the matrices (2), and the boundary conditions (3), a easy calculation shows that, on a boundary edge e: Metallic boundary condition:

• Absorbing boundary condition:

We point out that classical boundary numerical fluxes fall in this class for some particular choices of 7 [PIP 99]. • Metallic boundary condition: with the mirror state technique, the flux is given by taking (5) with vr^e = [ —E Hx Hy ] (if the interior state is vie = [ E Hx Hy f):

which is exactly the previous one, for 7 =

Numerical analysis

• Absorbing boundary condition: the flux is given by taking v1ffe (no incoming waves):

129

— 0 in (5)

which is equivalent to the formulation above, only for the limit case 7 = 0.

4. Main Results Under some regularity assumptions on the MO, the Ai and £7, (1) admits a unique solution u <E V = C°([0, T\, (H1^))™) n C^QO, T], (£ 2 (Q)) m )

[RAU85]. Theorem 1 (Comparison) For T > 0, suppose that v € £ 2 (]0, T[xQ) 25 suc/i that • there exists ^ £ V (the dual space of V) such that Vip € V,

• there exists a measure v such that

then with Theorem 2 (Convergence Rates) Under the following CFL conditions: • on the interior interfaces,

• on the boundary edges,

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Finite volumes for complex applications

the approximate solution v^ given by the finite volumes scheme (4) converges to the exact solution u of (1), and we have

5. Proofs

5.1 The Theorem of Comparison An easy calculation yields, for any £ C1([0, T] x fi) such that (j>(t, x] > 0 on]0, T[xfi,

The theorem 1 is obtained by taking (j)(t, x) = T — t (dt

for Complex Applications II

Sponsored by :

© HERMES Science Publications, Paris, 1999 HERMES Science Publications 8, quai du Marche-Neuf 75004 Paris Serveur web : http://www.hermes-science.com ISBN 2-7462-0057-0 Catalogage Electre-Bibliographie Finite Volumes for Complex Applications II — Problems and Perspectives Vilsmeier, Roland* Benkhaldoun, Fayssal* Hanel, Dieter Paris : Hermes Science Publications, 1999 ISBN 2-7462-0057-0 RAMEAU : elements finis, methode des analyse numerique DEWEY : 515 : Analyse mathematique Le Code de la propriete intellectuelle n'autorisant, aux termes de 1'article L. 122-5, d'une part, que les « copies ou reproductions strictement reservees a 1'usage prive du copiste et non destinees a une utilisation collective » et, d'autre part, que les analyses et les courtes citations dans un but d'exemple et d'illustration, « toute representation ou reproduction integrate, ou partielle, faite sans le consentement de 1'auteur ou de ses ayants droit ou ayants cause, est illicite » (article L. 122-4). Cette representation ou reproduction, par quelque precede que ce soit, constituerait done une contrefacon sanctionnee par les articles L. 335-2 et suivants du Code de la propriete intellectuelle.

Finite Volumes for Complex Applications II Problems and Perspectives

editors Roland Vilsmeier Fayssal Benkhaldoun Dieter Hanel

Second International Symposium on Finites Volumes for Complex Applications Problems and Perspectives July 19-22, 1999, Duisburg, Germany Internet adress : http://www.vug.uni-duisburg.de/FVCAII/contents.html

Organizing Institutions Institut fur Verbrennung und Gasdynamik (IVG), University Duisburg, Germany INSA de Rouen, France

Scientific Committee F. Benkhaldoun, LMI, INSA de Rouen, France R. Borghi ESM2, IMT-Technopole, Marseille, France A. Dervieux, INRIA Sophia Antipolis, France T. Gallouet, Universite Aix-Marseille I, France D. Hanel, IVG, University Duisburg, Germany D. Kroner, Institute f. Angewandte Mathematik, University Freiburg, Germany I. Toumi, CEA, Saclay, France J.-R Vila, INSA de Toulouse, France R. Vilsmeier, IVG, University Duisburg, Germany N. R Weatherill, University of Swansea, UK G. Wittum, IWR, University Heidelberg, Germany

Invited Keynote Lectures R. Abgrall, Universite de Bordeaux 1, France F. Coquel, CNRS, Paris, France G. Degrez, von Karman Institute, St-Genesius-Rode, Belgium R. Klein, Konrad-Zuse-Zentrum f. Informationstechnik, Berlin, Germany R. Lazarov, Lawrence Livermore National Laboratory, USA J. M. Ghidaglia, ENS de Cachan, France S. Noelle, Institute f. Angew. Mathematik, University Bonn, Germany

Contents

Editors preface

XIII

Invited speakers

1

Construction of some genuinely multidimensional upwind distributive schemes — R. ABGRALL

3

A Roe-type Linearization for the Euler Equations for Weakly Ionized Gases

F. COQUEL, C. MARMIGNON

11

Multidimensional Upwind Residual Distribution Schemes and Application

H. DECONINCK, G. DEGREZ

27

Overcoming mass losses in Level Set-based interface tracking schemes

Th. SCHNEIDER, R. KLEIN

41

Coupling mixed and finite volume discretizations of convection-diffusionreaction equations on non-matching grids R.D. LAZAROV, J.E. PASCIAK, P.S. VASSILEVSKI

51

Numerical computation of 3D two phase flow by finite volumes methods using flux schemes — J. M. GHIDAGL1A

69

The MoT-ICE : a new high-resolution wave-propagation algorithm based on Fey's Method of Transport — S. NOELLE

95

Numerical Analysis

115

Error estimate for a finite volume scheme on a MAC mesh for the Stokes problem — P. BLANC

117

Convergence Rate of the Finite Volume Time-explicit Upwind Schemes for the Maxwell System on a Bounded domain Y. COUDIERE, P. VlLLEDEU

125

VI

Finite volumes for complex applications

Flux vector splitting and stationary contact discontinuity — F. DUBOIS

133

Analysis of a Finite Volume Solver for Maxwell's Equations F. EDELVIK

141

A result of convergence and error estimate of an approximate gradient for elliptic problems — R. EYMARD, T. GALLOUET, R. HERBIN

149

Finite volume approximation of elliptic problems with irregular data T. GALLOUET, R. HERBIN

155

Analysis of a finite volume scheme for reactive fluid flow problems A. HOLSTAD, I. LIE

163

Convergence of a finite volume scheme for a nonlinear convectiondiffusion problem — A. MICHEL

173

Convergence analysis of a cell-centered FVM H.P. SCHEFFLER, R. VANSELOW

181

Error estimates on the approximate finite volume solution of convection diffusion equations with boundary conditions T. GALLOUET, R. HERBIN, M. H. VIGNAL

189

The limited analysis in finite elasticity — LA. BRIGADNOV

197

Entropy consistent finite volume schemes for the thin film equation G. GRUN, M. RUMPF

205

Convergence of finite volume methods on general meshes for non smooth solution of elliptic problems with cracks P. ANGOT, T. GALLOUET, R. HERBIN

215

Application and analysis of finite volume upwind stabilizations for the steady-state incompressible Navier-Stokes equations — L. ANGERMANN

223

A new cement to glue non-conforming grids with Robin interface conditions Y. ACHDOU, C. JAPHET, F. NATAF, Y. MADAY

231

Finite Volume Box Schemes — J.-P. CROISILLE

239

On nonlinear stability analysis for finite volume schemes, plane wave instability and carbuncle phenomena explanation — M. ABOUZIAROV

247

Innovative schemes

253

A comparison between upwind and multidimensional upwind schemes for unsteady flow — P. BRUFAU, P. GARCIA-NAVARRO

255

Reformulation of the unstructured staggered mesh method as a classic finite volume method — B. PEROT, X. ZHANG

263

A mixed FE-FV algorithm in non-linear solid dynamics — S.V. POTAPOV

271

Contents

VII

An Euler Code that can compute Potential Flow — M. RAD, P. ROE

279

Finite volume evolution Galerkin methods for multidimensional hyberbolic problems — M. LUKACOVA-MEDVIDOVA, K.W. MORTON, G. WARNECKE

289

Nonlinear anisotropic artificial dissipation - Characteristic filters for computation of the Euler equations T. GRAHS, A. MEISTER, T. SONAR

297

Nonlinear projection methods for multi-entropies Navier-Stokes systems C. BERTHON, F. COQUEL

307

About a Parallel Multidimensional Upwind Solver for LES D. CARAENI, S. CONWAY, L. FUCHS

315

A higher-order-accurate upwind method for 2D compressible flows on cell-vertex unstructured grids — L. CATALANO

323

A New Upwind Least Squares Finite Difference Scheme (LSFD-U) for Euler Equations of Gas Dynamics — N. BALAKRISHNAN, C. PRAVEEN

331

A finite-volume algorithm for all speed flows F. MOUKALLED, M. DARWISH

339

Preserving Vorticity in Finite-Volume Schemes — P. ROE, B. MORTON

347

On Uniformly Accurate Upwinding for Hyperbolic Systems with Relaxation — J. HITTINGER, P. ROE

357

Implicit Finite Volume approximation of incompressible multi-phase flows using an original One Cell Local Multigrid method S. VINCENT, J.P. CALTAGIRONE

367

New classes of Integration Formulas for CVFEM Discretization of ConvectionDiffusion Problems — E. P. SHURINA, T.V. VOITOVICH

377

Fields of application

385

Analysis of Finite Volume Schemes for Two-Phase Flow in Porous Media on Unstructured Grids — M. AFIF, B. AMAZIANE

387

A preconditioned finite volume scheme for the simulation of equilibrium two-phase flows — S. CLERC

395

Transient flows in natural valleys computed on topography-adapted mesh S. SOARES FRAZAO, J. LAU MAN WAI, Y. ZECH

403

A mixed Finite Volume/Finite Element method applied to combustion in multiphase medium — N. GUNSKY-OLIVIER, E. SCHALL

411

Turbulence Modeling for Separated Flows — L.J. LENKE, H. SMON

419

VIII

Finite volumes for complex applications

Simulation of unsteady Flow in a Vortex-Shedding Flowmeter S. PERPEET, A. ZACHCIAL, E. VON LAV ANTE

429

A Finite Volume Scheme for the Two-Scale Mathematical Modelling of TiC Ignition Process — A. AouFi, V. ROSENBAND

437

Two Perturbation Methods to Upwind the Jacobian Matrix of Two-Fluid Flow Models — A. KUMBARO, I. TOUMI, J. CORTES

445

Finite volumes simulations in magnetohydrodynamics — M. HUGHES, L. LEBOUCHER, V. BOJAREVICS, K. PERICLEOUS, M. CROSS

453

Finite Volume Method for Large Deformation with Linear Hypoelastic Materials — K. MANEERATANA, A. IVANKOVIC

459

A finite volume formulation for fluid-structure interaction C.J GREENSHIELDS, H.G. WELLER, A. IVANKOVIC

467

Boundary Conditions for Suspended Sediment V. BOVOLIN, L. TAGLIALATELA

475

Second order corrections to the finite volume upwind scheme for the 2D Maxwell equations — B. BlDEGARAY, J.-M. GHIDAGLIA

483

A MHD-Simulation in the Solar Physics A. DEDNER, C. ROHDE, M. WESENBERG

491

A Zooming Technique for Wind Transport of Air Pollution P.J.F. BERKVENS, M.A. BOTCHEV, W.M. LIOEN, J.G. VERWER

499

Computational Solid Mechanics using a Vertex-based Finite Volume Method G.A. TAYLOR, C. BAILEY, M. CROSS

507

Control volumes technique applied to gas dynamical problems in underground mines — E. VLASSEVA

517

Simulation of salt-fresh water interface in costal aquifers using a finite volume scheme on unstructured meshes — B. BOUZOUF, D. OUAZAR, I. ELMAHI

525

Progress in the flow simulation of high voltage circuit breakers X. YE, L. MULLER, K. KALTENEGGER, J. STECHBARTH

533

River valley flooding simulation — F. ALCRUDO

543

Modelling vehicular traffic flow on networks using macroscopic models

J.P. LEBACQUE, M.M. KHOSHYARAN

551

Finite Volume method applied to a solid/liquid phase change problem M. ELGANAOUI, P. BONTOUX, O. MAZHOROVA

559

Integrating finite volume based structural analysis procedures with CFD software to analyse fluid structure interactions M.A. WHEEL, A. OLDROYD, T.J. SCANLON, P. WENKE

567

Contents

IX

A generalized parcel method for the spray dispersion computation

B.NKONGA

575

Finite Volume Methods for Multiphysics Problems — C. BAILEY, M. CROSS, K. PERICLEOUS, G.A. TAYLOR, N. CROFT, D. WHEELER, H. Lu

585

A Finite Volume discretization and multigrid solver for steady viscoelastic fluid flows — H. AL MOATASSIME, S. RAGHAY, A. HAKIM

595

Complexity, Performance and Informatics

605

Various CG-type methods applied to finite volume schemes O. SCHMID, A. BUBMANN, E. VON LAV ANTE, M. MOCZALA

607

A Newton-Relaxation Finite Volume Scheme for Simulation of Dynamic Motion — B.A. JOLLY, M. RlZK

615

An Attempt to Develop a Multi Purpose FAS Multigrid Algorithm L. FOURNIER, O. GLOTH

623

On Higher Order Accurate Implicit Time Advancing for Stiff Flow Problems C. VlOZAT, E. SCHALL, A. DERVffiUX, D. LESERVOISIER

631

Numerical Solution of Steady 2D and 3D Impinging Jet Flows K. KOZEL, P. LOUDA, J. PRIHODA

639

Triangular, Dual and Barycentric Finite Volumes in Fluid Dynamics J. FELCMAN, M. FEISTAUER

647

Concepts for parallel numerical solution of PDEs — G. BERTI

655

Performing parallel direct numerical simulation of two dimensional heated jets S. BENAZZOUZ, V.G. CHAPIN, P. CHASSAING

663

Two-Dimensional Riemann Problems. Assessment Tests for Upwind Methods for Multi-Dimensional Supersonic Flow Problems J. VAN KEUK, J. BALLMANN

671

Robustness and accuracy on unstructured grids. Numerical experiments on finite volume schemes — E.A. MEESE, S.E. HAALAND

683

A validation of an efficient numerical method for 3-D complex flows

E.A. FADLUN, S. LEONARDI, R. VERZICCO, P. ORLANDI

693

Comparison of Two Finite Volume Methods for 3D Transonic Flows through Axial Cascades — J. FORT, J. FORST, J. HALAMA, K. KOZEL

701

An efficient and universal numerical treatment of source terms in turbulence modelling — B. MERCI, J. STEELANT, E. DICK

709

Comparison of numerical solvers for a multicomponent, turbulent flow E. XEUXET, A. FORESTIER, J.M. HERARD

717

X

Finite volumes for complex applications

Parallel Overlapping Mesh Technique for Compressible Flows J. ROKICKI, D. DRIKAKIS, J. MAJEWSKI, J. ZOLTAK

725

A comparison of Finite Volume and Higher-Order Finite Difference Schemes for the Solution of the Navier-Stokes and Euler equations M. MEINKE, Th. RISTER, R. EWERT

733

Simulation of 3D turbulent flow through steam-turbine control valves B.N. AGAPHONOV, V.D. GORYACHEV, V.G. KOLYVANOV, V.V. Ris, E.M. SMIRNOV, D.K. ZAITSEV

743

Adaptivity, Tracking and Fitting

751

An Adaptive Hybrid Object-Oriented Code for CFD-Applications-Adhoc3D U. TREMEL, H. BLEECKE, G. BRENNER, G. GREINER

753

Adaptive mesh refinement for single and two phase flow problems in porous media — M. OHLBERGER

761

Parallel solution of hyperbolic PDEs with space-time adaptivity P. LOTSTEDT, S. SODERBERG

769

Dynamic mesh generation with grid quality preserving methods

A. WICK, F. THELE A Finite Volume Method for Steady Hyperbolic Equations M. J. BAINES, SJ. LEARY, M.E. HUBBARD

777 787

Moving grid technology for finite volume methods in gas dynamics B.N. AZARENOK, S.A. IVANENKO

795

Numerical Simulation of Lifted Turbulent Methane-Air Diffusion Flames M. CHEN, N. PETERS

803

The application of a conservative grid adaption technique to 1D unsteady problems — M. CASTRO-DIAZ, P. GARCIA-NAVARRO

809

Application of mesh adaptive techniques to mesh convergence in complex CFD D. LESERVOISIER, A. DERVIEUX, P.L. GEORGE, O. PENANHOAT

817

Multidimensional Fully Adaptive Finite Volume Schemes for the Numerical Simulation of Stiff Combustion Front Propagation in Condensed Phase A. AOUFI

825

Mathematical and numerical modeling of a two-phase flow by a Level Set method — S. ROUY, P. HELLUY

833

Multiresolution analysis on triangles: application to conservation laws

A. COHEN, S.M. KABER, M. POSTEL A local level set method for the treatment of discontinuities on unstructured grids — L. TRAN, R. VILSMEIER, D. HANEL

841 849

Contents

XI

Varia

857

A Stabilized Version of Wang's Partitioning Algorithm for Banded Linear Systems — V. PAVLOV

859

On Jeffreys Model of heat conduction — M. DRYJA, K. MOSZYNSKI

867

Investigation of some method for cavitating jet S. OCHERETYANY, V.V. PROKOFIEV

Index des auteurs

875

887

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Editors preface Finite Volume methods are methods directly related to the numerical solution of conservation laws. Systems of such conservation laws govern wide fields of physics and the efficiency of corresponding solution methods is an essential requirement from basic research and industry. Since the efficiency of any method must be measured by the quality of the result compared to the computational cost to spend for, corresponding developments are widely spread, ranging from very fundamental numerical analysis up to the efficient use of modern computer hardware. Although in the past the numerical methodology has made large progresses, many problems and difficulties remain, requirering further intensive research. New fields of application as well as the coupled simulation of different physical phenomena become accessible due to improved solution techniques and growing computer capacities. However, these new possibilities introduce again new physical and mathematical problems to be solved. The development of new methods as well as the extension of existing ones requires intensive and critical investigations and careful validation. One of the aims of this conference is therefore to bring together people working in theory and practice for fruitful and critical discussions about methods, their advantages and drawbacks and related experiences from arbitrary applications. The present proceedings summarize the contributions to be presented at the second international symposium on Finite Volumes for Complex Applications Problems and Perspectives. The first symposium of this series was held summer 1996 at INSA de Rouen in France. Based on the success of this first conference, the symposium in Duisburg has again received an unexpected high attention in the numeric community. After a critical review of the submitted contributions, 98 papers by authors from 20 countries are presented in this volume. In a rough estimation, about half of the contributions can be assigned to analysis and numerics of different methods whereas the other half is essentially concerned with application and computational aspects of methods. We would like to thank all persons, who contributed to the conference and to this book of proceedings. First of all, we want to mention all the authors as well as the other members of the scientific committee, for the work of writing the papers as well as selecting these in remarkably short times. We would like to extend our thanks, acknowledging the help from the numerical staff, secretaries and students at the IVG in Duisburg and at INSA de Rouen, keeping their good mood whenever. Finally we want to thank the following organizations for the financial support: Deutsche Forschungsgemeinschaft, Ministerium fur Wissenschaft und Forschung Nordrhein-Westfalen, Duisburger Universitatsgesellschaft, Hewlett Packard and Sun Microsystems.

Fayssal Benkhaldoun, Dieter Hanel and Roland Vilsmeier

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Construction of genuinely multidimensional upwind distributive schemes

Remi Abgrall Universite Bordeaux I Mathematiques Appliquees 351 Cours de la Liberation 33 405 Talence Cedex, France

ABSTRACT We present a construction of a class of upwind residual schemes for the Euler equations of fluid mechanics. It naturaly generalises the PSI scheme of Deconinck, Roe and Struijs. We show some of its theoretical properties. Numerical illustrations indicates obvious advantages over the now classical finite volumes upwind schemes. Key Words : Upwind schemes, Residual schemes, Unstructured meshes.

1. Introduction Most of the "modern" industrial codes that are used to simulate complex compressible fluids are written according to the ideas developed in the 80's by van Leer, Harten, Osher, Roe, and many others. These codes are versatile, robust and rather accurate. However, in some situations, the results are somewhat disapointing : the numerical viscosity of these schemes is still too high, despites some attemps to reduce it. It is still difficult to accurately compute the lift and drag of an airfoil. These schemes are also very mesh dependent : the formulation deeply rely on the shape of the control volume. Their definition is many times not related to the physics of the problem under consideration. For these reasons, since a few years, some researchers have tried to find alternative formulations of the problem that could - in principle - lead to less dissipative and mesh independant schemes. Among the first, one might quote [Da] who tried to represent the fluxes in term of Riemann problems in direction related to the fluid. A very promissing formulation is certainly that of upwind residual schemes, who were pioniered by P.L. Roe, the H. Deconinck and their coau-

4

Finite volumes for complex applications

thors. These schemes are also related to the SUPG schemes of Hughes, or the streamline diffusion method by Johnson. If the design principle of the upwind residual schemes are rather clear for scalar convection equations, this is not anymore true for systems, and in particular the Euler equations of fluid mechanics. The aim of the present paper is to give the status of a research toward that goal. The paper is divided into four parts. In the first one, we recall the upwind residual formulation and the design principle for scalar equations. Then we reformulate the PSI scheme of Deconinck, Roe, Struijs and Sidilkover in a way that is easier to generalise in multiD. More precisely, it is seen as a blending between the N scheme and the LDA scheme. In the third part, we discuss design principle for the Euler equations, and propose a class of schemes. Then numerical examples are given. 2. Upwind Residual schemes 2.1 The Euler equations We consider the Euler equations for fluid mechanics with initial and boundary conditions

inflow and/or wall conditions. In (1), the vector of conserved variables is W — (p, pu, pv, E)T, the x-component of the flux is Fx = (pu,pu2 + p, puv, u(E + p))T. Its y-component has a similar expression. In the problem of interest, the pressure p is related to W via p = (7 _ ! ) ( £ _ _ p ( u 2 + v 2 )) with 7 = 1.4. The solutions of (1) has to fullfill £ the second law of thermodynamics : we have to have

where S = —ps (with s = log ( -^-)) is the mathematical entropy. In [Ta], E. Tadmor has shown that the solutions of (1-2) satisfy the following minimum principle

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2.2 Numerical schemes

2.2.1 Generalities The discretisation of (1) is carried out on a mesh made of triangles. The list of nodes is {Mi} i = 1 , n s . The generic name of a triangle is T, its vertices are denoted by Mi1; Mi2, Mi3, or 1, 2, 3 when there is no ambiguity. The schemes for (1) are written as

In (4), W™ is an approximation for W(Mi,nA£), \Ci\ is the area of the dual cell associated to node Mi1 The residual $iT must satisfy

In this conservation relation Fh is an approximation of F that has to be continuous across the edges of the triangles. Under classical asssumptions, we have a Lax Wendroff-like theorem [AMN] : the scheme, if it converges, converges to weak solutions of (1). In this paper, we follow the approach of Roe-DeconinckStruijs [SDR] via the parameter vector Z — (^/p, <

where 2Ki = ( n i ) x A + (ni) y B. One can show that Ki is diagonalisable and has real eigenvalues.

2.2.2 Design principles The schemes are constructed follwing three design principles - the scheme is upwind : if Ki only has negative eigenvalues, $J = 0. ^n particular, we have |Ci| = \ ^T M - P T l^l-

6

Finite volumes for complex applications

- the scheme must be linear preserving : if $T = 0 then

In the definition of these schemes, the matrix N appears. It is the inverse of X]i=i ^"i~- This matrix is not always invertible. However, for the Euler equations, one can show [AMN] that it is always invertible except at stagnation points. In any case, one can always give a meaning to K^Z or N$T because the Euler equations are symetrizable, see [AMN]. Hence, there is no problem in the definition of ^f7 or $fDA. Both schemes are clearly upwind. The LDA scheme is linear preserving contrarily to the N scheme. The N scheme is monotone. This is very obvious for its scalar version because in that case we have

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with Cij > 0. The matrix equivalent formulation involve terms like K^NK~ that are difficult to handle. In all numerical experiments (with a large variety of geometries and flow conditions) seem to indicate that the system N scheme also satisfies a discrete version of (3). Barth [Ba] has shown that the N scheme, for a linear symetric hyperbolic system is locally dissipative. In [Ab], we show that the LDA scheme is also locally dissipative for a linear symetric hyperbolic system. 3. A positive linear preserving scheme for scalar equation : the PSI scheme revisited We consider the scalar versions of the N and LDA schemes and the following blended scheme The firt remark is that this scheme satisfies the conservation constraint (5) whatever / e R. This scheme is upwind by construction since the N and LDA schemes are upwind. We now consider the positivity issue with the same tech/

nique as D. Sidilkover, $< = l^

&LDA\

+ (l-l)^DA = (/ + (!- O-^T ) • Thanks V

^i

/ If we set

the positivity is obtained if

a solution is given by

where 0 and

T

else. Simple algebraic manipulations r —I shows that if / is chosen with the = sign in (7), the scheme is positive and linear preserving. In fact, it is identical to the PSI scheme. 4. A scheme for the Euler equations Following the same ideas, we consider a scheme written as

where 1 is a matrix. In order to illustrate the design principles, we consider 1 = 1 = Id where / 6 M, but a more sophisticated method is developped in [Ab].

8

Finite volumes for complex applications Conditions Top Bottom

P 1.4 0.7

P 1 0.25

Mach number 2.4 4

TAB. 1: Conditions for the interaction of 2 parallel supersonic flows Let us denote by

and thanks to the numerical experiments, we assume that the system N scheme has a local minimum principle for the specific entropy. Following the same arguments as for the scalar PSI scheme, the conditions are

where here

One can show that this scheme is also locally dissi-

pative[Ab]. 5. Numerical experiments

We present some results obtained in two different test cases. The first one is the iteraction of two parallel supersonic flows. The conditions are given in Table 1 The mesh is given on Figure 1-a The isovalues of the density are presented on Figure 1-b. A very clear improvement of the results can be observed. The new scheme give monotone results that are more accurate than those of the finite volume scheme (MUSCL extrapolation on conserved variables). The second test case is a GAMM test case : Naca0012, Mach number : 0.85, angle of incidence : 1 degree. We show the isolines of the Mach number (Figure 2-a) and the isoline of the reduced entropy (Figure 2-b). It is clear that the slip line out of the leading edge is improved as well as the entropy profiles. 6. Conclusions

We have sketched the construction of upwind residual schemes that are also linear preserving. Some numerical example indicate that these new schemes are more accurate than the now classical finite volume schemes.

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Bibliography

[Ab]

R. ABGRALL. Upwind residual schemes on unstructured meshes, in preparation.

[AMN]

R. ABGRALL, K. MER, AND B. NKONGA. A Lax-Wendroff type theorem for residual schemes. In M. Hafez and J.J. Chattot, editors, Proceeding of a conference for P.L. Roe's 60th birthsday. Wiley, to appear.. T.J. BARTH. Some working note on the n scheme. Private communication, 1996. S.F. DAVIS. A rotationaly based upwind difference scheme for the Euler equations. J. Comp. Phys., 56 :65-92, 1983. R. STRUIJS, H. DECONINCK, AND P. L. ROE. Fluctuation splitting schemes for the 2d euler equations. VKILS 1991-01, Computational Fluid Dynamics, 1991.

[Ba] [Da] [SDR]

[Ta]

[DvW]

E. TADMOR. The numerical viscosity of entropy stable schemes for systems of conservation laws. Math. Comp., 49 :91-103, 1987. E. VAN DER WEIDE AND H. DECONINCK. Positive matrix distribution schemes for hyperbolic systems. In Computational Fluid Dynamics '96, pages 747-753. Wiley, 1996.

FIG. 1: (a)-Mesh for the interaction of 2 parallel supersonic flows ; (b)- Density isolines for the supersonic flows. Top-left : N scheme p G [0.7,1.4], top-right : second order MUSCL finite volume scheme p 6 [0.689,1.403] , bottom-left : LDA scheme p 6 [0.615,1.427], bottom-right: present scheme p € [0.698,1.402]

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Finite volumes for complex applications

FIG. 2: (a)-Mach number isolines for the Naca0012 test case. Top-left : N scheme M G [0.05,1.394] , top-right : second order MUSCL finite volume scheme M <E [0.05,1.422], bottom-left : LDA scheme M 6 [0.05,1.425], bottom-right : present scheme M e [0.04,1.522]; (b)-Reduced entropy £ = •f IN scheme S G [0,0.038],second order MUSCL finite volume scheme S°°e [-0.009,0.039], LDA scheme £ € [-0.009,0.091], present scheme S e [-0.0003,0.032]

A Roe-type Linearization for the Euler Equations for Weakly Ionized Gases

Frederic COQUEL LAN CNRS Tour 55-65 5ieme, U.P.M.C. 75252 Pans Cedex 05 Claude MARMIGNON ONERA, BP 72 92322 Chatillon Cedex

ABSTRACT This paper is devoted to the numerical approximation of the discontinuous solutions of the Euler equations for weakly ionized mixtures of reacting gases. The main difficulty stems from the non conservative formulation of these equations due to a widely used simplifying assumption. We show how to derive a well-posed conservative reformulation of the equations from the analysis of the associated connective-diffusive system. We then propose an exact Roe-type linearization for the equivalent system of conservation laws. Our results can be seen as an extension of the classical Roe average, for nonlinearities that cannot be recast under quadratic form. Key Words: Convective-diffusive systems. Nonlinear hyperbolic systems. Non conservative products. Shock solutions. Roe-type linearization.

1. Introduction

This work treats the numerical approximation of the solutions of a convectivediffusive system, we write for short as

This system governs ionized mixtures of reacting gases in thermal nonequilibrium. Such plasma are studied here in the context of large Mach number flows.

12

Finite volumes for complex applications

The solutions we are interested in, are thus mainly driven by the underlying first order system. The main properties of the extracted first order system

are reported below. This nonlinear system will be seen to be hyperbolic so that its solutions are known to develop, generally speaking, discontinuities in a finite time. But when dealing with discontinuous solutions of (2), a major difficulty arises : there does not exist a flux function, say f, such that A(u) = Vuf (u). In other words, the hyperbolic system (2) is under non conservation form. It is known that the non conservative products involved in A(\i)dxu have no classical sense at the location of a shock since they cannot be given a unique definition within the standard framework of distributions. For this reason, it must be recognized that an additional information is required in order to specify the definition, e.g. the value, of the non conservative product A(u)dxu at shocks. This difficulty has motivated some recent works. We refer in particular to the work by LeFloch [8], DalMaso-LeFloch-Murat [6] where non conservative products are defined on the basis of a fixed family of paths <]? in the phase space : After LeFloch [8] and Sainsaulieu [11], the choice of a particular family of paths $ is dictated by the additional informations brought by the full second order convective-diffusive system (1) (see below for a brief survey). The key feature is that the definition of shock solutions heavily depends on the shape of the diffusive tensor £>(u) which is modeled in agreement with the physics. These definitions provide us with a relevant setting for defining the discontinuous solutions of the non conservative hyperbolic system (2). Once defined, the first order system is well-posed and its numerical approximation could be tackled. However, two difficulties arise in that way. First, a close formula for shock solutions is in general not available. Furthermore, even when explicitely available, we have illustrated [4] that the error in the discrete capture of shock solutions unacceptably grows with the strenght of the shock. We refer to LeFloch-Liu [9] for an error analysis devoted to the scalar case. At this stage, these two difficulties make the numerical approximation of the (strongly) discontinous solutions of (3) to be virtually untractable. To overcome these two difficulties on the same time, we propose to study the existence of a conservative formulation for system (3) that is compatible with the diffusive tensor V. That is to say, we ask for the existence of (at least) one change of variables v = v(u) that brings the non conservative second order

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system (1) (with c«;(ii) = 0) to a fully conservative convective-diffusive system

Let us emphasize that not only the first order system must find a conservation form (PI) but also that the second order operator must stay under conservation form (P2). Actually, these two requirements (P1),(P2) ensure the equivalence of the shock solutions of (3) where <£ and T> are compatible and the shock solutions of We refer to Sainsaulieu-Raviart [12] for a proof. The benefit of such an equivalence is twofold. In a first hand, the shock solutions of (3) are now explicitely given by the Rankine-Hugoniot jump relations associated with (5). In a second hand, Riemann solvers under conservation form can be applied to (5) in order to approximate the equivalent weak solutions of (3). As reported below, a specific family of change of variables turns out to fulfill both (PI) and (P2). For the associated equivalent systems of conservation laws, we then show how to derive an exact Roe-type linearization. 2. Analysis of the extracted first order system

In this section, we focus ourselves on the definition of the extracted first order system (2), the precise shape of the diffusive tensor 1) will be discussed later on. We treat mixture of gases made of electrons and n heavy species, ni, 1 < ni < n of them being ionized. All the species we consider are described with the same mean velocity v. To account for the smallness of the mass ratios Me/M2- « 1, z'G {!,..., n}, the electron gas is endowed with a temperature Te distinct from the temperature T of the heavy species mixture. Moreover, nv, I < nv < n, molecular species have their own vibrational temperature Tvj, j G {1, • • • , nv}. Neglecting at this stage the thermo-chemical relaxation terms, the extracted first order system (2) writes :

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Finite volumes for complex applications

The main properties of (6) stated in the next section will enlighten several relationships with the usual Euler system. Here, Y,, i 6 {1, ...,n} denotes the ith species mass fraction. The conservation laws (6b),(6c) respectively govern the momentum and total energy E of the mixture of the heavy species plus the electron gas. Conservation laws (6e) refer to the vibrational energies pYjevj of the nv molecular species that are assumed to be in thermal nonequilibrium. These (n + nv + 2) conservation laws are supplemented by a balance equation (6d) : the expected conservation law for the electron gas energy pYeEe must be, in fact, balanced by the work of the electric field £. The required closure equations are as follow. The electron gas pressure pe obeys : while the pressure law p for the mixture of heavy species is defined by :

with 7tr G ]l,3j. Here, Cj-(T) refers to the energy of the internal modes of species i at equilibrium with the temperature T and h® denotes its heat of formation. The last closure equation to be specified deals with the electric field £. The expected closure equation for £ should be the Poisson law, written here in a dimensionless form :

In (9), € refers to a parameter proportional to the Debye lenght. This parameter actually yields a rough estimate of the spatial resolution that is required to approximate (9) : namely O ( c — l ) . However in our applications, this parameter turns out to be extremely small (see [14] for instance) To make the numerical approximation tractable, we are lead to let c goes to zero. Doing so, the Poisson law degenerates to the so-called local charge neutrality condition while the closure equation for E is classically given by :

Here, z,- refers to the electric charge of the ith ionized species. The stricking feature of the closure equation (10) stays in that it involves partial derivatives

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and in particular makes £ to be, generally speaking, a measure at the discontinuities of u ! The work of the electric field Neqe£ x v in (6d) is clearly a product involving first order and zero order terms and furthermore, it cannot be put under divergence form. Let us stress that the product Neqe£ x v cannot be understood, as it is commonly handled (see [2] for instance), as a source term, e.g. a function depending solely on zero order terms. On the contrary, such a product is a constitutive part of the first order system.

2.1. Basic properties of the extracted first order system In this section, we state some of the main properties of the first order system (6). The results given below intend to provide a deeper insight into the system under consideration and in particular to shed light on its relationships with the standard 3x3 Euler equations. The phase space Q associated with (6) is the following subset of Rp, p = n + nv + 3 :

We begin with the following result devoted to the smooth solutions of (6). Lemma 1. Let u : R x R+ —>• Q be a Cl solution of (6). Then, the closure equation on the electric field £ reads :

Furthermore, u satisfies the following balance equations in non conservation form :

Dropping the electron gas pressure pe in (13c), the three equations (13b) to (13d) are easily recognized to coincide with the ones governing the smooth solutions of the 3x3 Euler system. Here, the heavy species mixture and the electron gas equally contribute to the velocity balance equation : each with

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Finite volumes for complex applications

their own pressure. The pressure balance equations (13d) and (13c) read the same but the key point stays in their complete decoupling. We indeed have : Proposition 2. i) The first order system (6) is hyperbolic in the phase space Q and A(u) : Q —> Mat(Rp} admits the following three real eigenvalues :

where the eigenvalue v has an order of multiplicity (n + nv + 1). 11) Under standard thermodynamic assumptions, the fields associated with the eigenvalues v ± c are genuinely nonlinear. Hi) The fields associated with the eigenvalue v are linearly degenerate. Their Riemann invariants are given by v and (p + pe). iv) The closure equation (12) for Neqe£ never involves a product of a Dirac mass against an Heaviside function and is thus well-defined. v) The non conservative product Neqe£ x v has no classical sense for the shock solutions of (6). According to the statement v), the shock solutions of (6) stay unknown within the standard framework of distributions. The nonconservative products involved in A(u)dxvi can nevertheless receive a definition in the recent setting proposed in [6]. Indeed, such products can be defined thanks to a given fixed family of paths, denoted $, that are subject to some consistency and smoothness conditions (see [6] for the details). For a shock discontinuity separating the two states u/, and u#, the non conservative product is defined by :

Note that since A is not a jacobian matrix, the definition (16) entirely depends on the choice of <£. After LeFloch [8] and Sainsaulieu [11], the relevant choice of <£ comes from the study of the full second order convective diffusive system (1). Indeed, the shock solutions of (6) can be defined as the limit when the diffusion is neglected of the travelling waves solutions of (1). More precisely, a function u : R x jR+ —>• £1 is a travelling wave solution of (1) connecting the states u/, and UR with speed

Next let be given a small parameter e > 0 and let us consider the function

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u(x f a t } - It can be easily seen that this function is also a travelling wave solution but of the convective-diffusive system (1) with vanishing viscosity :

Furthermore, since d^ue £ L^(R) with ||c/£Ue||£i = ||c^u||£i, we deduce that. {ue} tends almost everywhere as e goes to zero to the step function u := U£-f(u/£ — U L ) H ( 0 ) while for any given e > 0, the followingjump like conditions hold true :

Motivated by the above calculations, LeFloch [8] (see also Sainsaulieu [11]) has defined the discontinuous step function u to be a shock solution of the first order extracted system (2) which is compatible with the diffusive tensor V. It is essential to notice that by contrast with conservative hyperbolic systems, the travelling wave solutions and therefore the step function u depend on the shape of D. Indeed, two non proportional diffusive tensor generally yield distinct shock solutions for the underlying convective system. Such an issue is actually illustrated by the Proposition 8 stated below. The first order extracted system (6) can be closed according to the above framework and its numerical approximation could be tackled. However and as explained in the Introduction, two difficulties arise when dealing with the numerical approximation of the discontinuous solutions of (6). In order to circumvent these two difficulties simultaneously, we have put forward the need for admissible change of varaibles that recast (6) in the full conservation form (4). We indeed have : Proposition 5 (Sainsaulieu-Raviart [12]). Let be given a C1 diffeomorphism W : Q —> £1. Assume that for any given smooth solution u : R x R+ —>• £7 of the non conservative convective diffusive system :

the change of variables v = ^(u) yields a solution of the following SLC :

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Finite volumes for complex applications

Then, the shock solutions of the extracted first order system of (21), e.g. the ones compatible with D, do coincide with the shock solutions of the extracted conservative first order system of (22) :

The benefit of this equivalence is twofold : first the shock wave solutions to (6) are now explicitely given by the Rankine-Hugoniot relations associated with

and in a second hand, upwind methods in conservation form can be applied to (24) in order to approximate the equivalent weak solutions of (6) that are compatible with D. Motivated by these strong benefits, we have laid in a related work [3] the foundations for a systematic characterization of all the CldifFeomorphisms ^ that bring (22) from (21). The starting point stems from the following observation : the equivalence stated in (23) requires in fact the fulfilment of two distinct conditions. Namely, not only the first order system must meet a conservation form (P1) but also that the second order operator must stay under conservation form (P2). In what follows, we study for existence the admissible changes of variables ^ that satisfy both requirement (P1) and (P2). 2.2.1. Fulfilling (P1). In order to exhibit the C 1 -diffeomorphismsthat satisfy the requirement (P1), we propose to characterize in a first step all the additional scalar conservation laws satisfied by the smooth solutions of (6). Each additional conservation law indeed yields an obvious change of variables by substitution with the single equation in non conservation form (6d). We specifically have : Theorem 6 Let u : R x /?+ —>• Q be a Cl solution of (6). Then u satisfies all the additional (non trivial) scalar conservation laws :

where g : £1 —)• R denotes an arbitrary (say of class C1) function of the common Riemann invariants of the two genuinely nonlinear fields : e.g.

Assume that

d Q

? Te ^ 0, then the Cl diffeomorphismm^g

M/ p (u) = ({pYj}, pv, pE, pg(\i), pYjeVtj)

: £2 —)• f2, u —>

yields the smooth solutions of (6)

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to be equivalent smooth solutions of the following hyperbolic system in conservation form :

The above family of C^diffeomorphisms that satisfy (PI) for the smooth solutions of (6), might look rather large. However, the next statement shows that all these possible candidates for the fulfilment of both (PI) and (P2) are indeed closely related. Corollary 7. i) All the additional conservation laws specified in (26) can be recovered from the following finite set of transport equations (since dtg(u) + vdxg(u) = 0;

n) The functions g : Q —>• jR are Riemann invariants for the two genuinely nonlinear fields and thus stay constant across the associated rarefaction waves, iii) Assume that one of the change of variables ^g specified in (28) meets the requirement (P2). Then, necessarily the related function g(u) stays continuous for (so constant across) the shock solutions of (6) that are compatible with D. The consequences of properties i) to iii) are briefly discussed in the next section since they will serve as a natural guideline for investigating for validity (P2) with all the possible candidates Wg. If one of them turns out to satisfy (P2), the resulting conservative system (28) and the extracted first order system (6) will not only describe the same smooth solutions but will also admit the same discontinuous solutions : namely, (24) will hold true. This issue is addressed below. 2.2.3. Fulfilling (P2). The existence of a C 1 -diffeomorphism that obeys (P2) obviously heavily depends on the shape of D. In view of iii), Corollary 7, its shape must be compatible with the conservation across the shock-solutions characterized by

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D of a least one function g(u) in the form (26). According to the Definition 4, the characterization i), Corollary 7, leads us to derive the travelling wave equations that govern (Yi, in(p//?7), ln(pe/p~is), Yjevj) in the presence of V and to study the nonlinear combinations in the form (26) that yield a fully conservative equation. The analysis [4] shows that none of the \&g given in (27) can achieve such a requirement for a general diffusive tensor D. However, this negative result can be bypassed provided that a physical assumption is made. This is the matter of the next statement. Proposition 8. i) Let the diffusive tensor D(u) be defined by

where v, K respectively denote the viscosity and the thermal conductivity of the heavy species mixture, Ke the thermal conductivity of the electron gas and KVJ the one of the jth molecular species in vibrational nonequilibrium. Then, the associated travelling wave u satisfies for all t; £ R, with M = p(£)( v (£) — cr) =

Furthermore, for arbitrary (*/, K, Ke, KV,J], none of the Cl-diffeormophisms specified in (27) does satisfy the property (P2). ii) Assume that Ke — 0, e.g. that the diffusive tensor £>(u) is now given by :

Then, all the Cl diffeomorphisms tfg based on 7e for the electron gas stays constant accross the shock waves that are compatible with (30). We underline that such a consequence is indeed compatible with the thermodynamic second principle : the heavy species specific entropy p/p^ indeed strictly increases accross shock waves (see also n) below). The physical interpretation of these admissible Cldiffeomorphisms can be found in Zeldovich-Raizer [14] (vol. 2). Proposition 8 provides us with a whole family of \&g that fulfill both (PI) and (P2).

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3. A Roe-type linearization

In this section, we propose a Roe method for the conservative hyperbolic system (33) in two space-variables. Since these equations stay invariant under rotations, the scheme makes use of the classical dimensional splitting and is thus derived for the associated quasi-lD system, we write with clear notations :

In what follows, we show how to derive an exact Roe-type linearization on the basis of an original Lemma for averagings. This Lemma will turn out to be central in the handling of the highly nonlinear pressure law pe. We just recall that a Roe-type linearization [10], we denote -B(v/,,V.R), has to satisfy the following three requirements:

In (37), the jacobian matrix V v ^ r (v): Q —> Mat(Rp} of the exact flux can be expressed in terms of the following set of arguments :

The precise shape of V v J r (v) is given in a companion paper [4]. Motivated by recent works [1], [13] devoted to mixtures of neutral gases, e.g. without ionization effects ; we seek for a linearization considering the following averaged form for V v ^ 7 (v) :

We have to define the set of averages involved in (39) so that the three conditions (37a)-(37c) hold true. Besides the hyperbolicity condition (37a), (37b) requires that such averages have to be derived so that the jump AJ r (v), which is of course nonlinear in v, can be reconstructed from the linearized form (39) of V v ^ r (v) times the jump Av. It is worthy to recall that for the 3x3 Euler system, the classical Roe average (see below) basically stems from the existence of a set of variables, the so-called parameter vector [10], for which all the

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nonlinearities involved in the system can be recast under quadratic form. Furthermore, within this framework, this Roe average turns out to be the unique average that yields (37a)-(37c). In our setting, some of the scalar conservation laws involved in (35)-(36) actually exhibit the same type of quadratic nonlinearities. We are thus lead to endow the related variables with the Roe averaging procedure : namely, we choose

But the pressure laws we have to deal with, bring a different type of nonlinearities. Averaging their partial derivatives indeed requires a specific treatment such that, besides (37a), the following consistency conditions hold true :

In fact, within the framework of neutral gases, Vvp has been already given suitable averaged forms. We refer in particular to the works by Abgrall [1] and Shuen-Liou-VanLeer [13]. When ionized species occur, we have shown [4] that the first identity in (41) can be derived from the framework of neutral gases provided that the second identity is valid. Our Roe-type linearization tacitly makes use of one of these relevant averages for V v p. From now on, we focus our attention on the derivation of an averaged form for V v p e which is consistent with the three requirements (37a)-(37b). In that aim, we state the following first result : Proposition 11. i) The averaged gradient V v p e can be recontructed from a set of three averaged partial derivatives :

where

if the

species i ionized and 0 otherwise, Next, assume that is associated with

and Vvp derived from [1] or [13]. Then, assumption (H) yields : ii) the Roe consistency condition (37b) is met iff for any given (v/,, v/?) £ ^2

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in) Whatever are these three averages, B(VL,VH] admits the set of eigenvalues (v — V c 2 , v, v + V c 2 ). Moreover, the real number c2 stays positive provided that for any given (v^, v#) G Q2, the following, where (7,7 e ) G ]1,3] 2 , fto/c/s £rue :

In order to enforce the validity of the two conditions (44) and (45), we now consider the following easy but key lemma for averagings (see also Abgrall [1], Godlewski-Raviart [15]). Averaging Lemma. Let be given (XL, XR), (yi, UR) two pairs of real numbers. Let (77,, TR) be any given pair of real numbers such that TL + TR ^ 0. Let us define the following unsymmetric r-averaging operators :

Then, the following identity holds true :

Let us apply for instance the above averaging Lemma to the pairs of interest (/>, X] with the £-averaging we define by gL=^/PL, QR — ^fpR• We easily get from (46) and (47) the well-known Roe identity, we write with classical notations :

Turning back to the general case, we emphasize that the identity (47) is indeed valid for any given pair (TL, TR). Taking advantage of such a degree of freedom, we introduce below a (wide !) family of unsymmetric averagings that makes always valid the required consistency condition (44). Equipped with these families, we next turn studying how to enforce the hyperbolicity condition (45). To that purpose, the specification of the underlying averaging operator is clearly the central issue. The statements, given below, summarize our main results. They are intended to shed some light on the application of the averaging Lemma we have introduced in order to enforce the validity of both (44) and (45).

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3.1. Enforcing the Roe consistency condition (44)In view of the electron gas pressure law (34), the required discrete identity (44) writes :

The above consistency condition involves increments in the conservative variables pYe, p and se while the underlying nonlinearities merely occur in terms of the primitive variables Ye, p and se/p. To make the needed linearization tractable, we specify in the next statement a convenient family of averages for (42) that shifts (44) from the conservative to the primitive variables. Proposition 12. Let us consider the following averaged forms for the partial derivatives (42) of the electron gas pressure law :

Then, for any given pair of states (VL, v#) ; the consistency condition (44) zs equivalent to the following identity, where p is defined in (48) :

In order to satisfy (51) and thus (44), it remains to define the unspecified mean values in (50). This is the matter of the next statement. Proposition 13. With the notations of the Averaging Lemma, let us define the averaged forms in (50), using respectively an arbitrary a-averaging operator :

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and an arbitrary (^-averaging operator :

Then, the identity (51) and thus the Roe consistency conditions (37b)~(37c) are always valid. In what follows, we shall assume for convenience that both a and (" stay non negative. Equipped with this wide family of relevant averagings, we now turn studying for validity the hyperbolicity condition (45). 3.2. Enforcing the hyperbolicity condition (45) The main result of this second step is as follows : Proposition 14. Let us respectively define the a and (, averaging operators in (52), (53) by

Then, (54), (55) provide us with the unique pair of averaging operators such that the hyperbolicity condition (45) holds true for any given (VL, v#) £ Q2. We have the following final statement. Theorem 15. Let us consider the following averaged form of V v ^ fr (v) :

with [YI, v, w, H, se/p, YjeVtj} given in (40), with Vvp derived from [1] or [13] and with V^pe constructed from (40), (50), (52)-(53). Then, for any given pair of states (VL, v#) € Q2, B(VL, VR) admits three real eigenvalues with a complete set of right eigenvectors. Moreover, this matrix satisfies the Roe consistency conditions (37b) and (37c). Therefore, B(VL, v#)

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is an exact Roe-type linearization for the system (35)-(36). We emphasize that the averaging operator built on (55) provides one with an extension of the classical Roe-average (48). Indeed, let us compare two of the nonlinearities that arise in the pressure laws : namely p7e x {h(^}} with P x (?(~) }• Now, it is sufficient to apply to the latter nonlinear expression our formula (55) with the associated exponent a = I in place of je to get v/p. [I]

R. ABGRALL, La Recherche Aerospatiale, vol 6, 31-43, 1988.

[2]

G. CANDLER AND R.W. MACCORMACK, AIAA paper 88-0511, 1988.

[4]

F. COQUEL AND C. M A R M I G N O N , Work in preparation.

[5]

S. CORDIER ET a/., Asymptotic Analysis, vol 10, 1995.

[6]

G. DALMASO, P.G. LEFLOCH AND F. MURAT, J. Math. Pure Appl., vol 74, 483-548, 1995.

[7]

B. LARROUTUROU, Computational methods in applied sciences, ECCOMAS, Eds Ch. Hirsch, Elsevier, 1992.

[8]

P.G. LEFLOCH, IMA Preprint series No 593, University of Minnesota, 1989.

[9]

P.G. LEFLOCH AND J. LIU, Math, of Comp., 1994.

[10]

P.L. ROE, J. Comp. Phys., 357-372, 1981.

[II]

L. SAINSEAULIEU, SIAM J. Appl. Anal., 1995.

[12]

P.A. RAVIART AND L. SAINSEAULIEU, Mathematical Methods and Models in Applied sciences, 1995.

[13]

J.S. SHUEN ET a/., NASA TM 100856, 1988.

[14]

YA. ZEL'DOVICH AND Yu. P. RAIZER, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomen, vol 1 and 2, Academic Press, 1966.

[15]

E. GODLEWSKI AND P. A. RAVIART, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer (Eds), Applied Mathematical Sciences, vol. 118, 1996.

Multidimensional upwind residual distribution schemes and applications

H. Deconinck and G. Degrez von Karman Institute for Fluid Dynamics Sint-Genesius-Rode, Belgium

ABSTRACT A review is made of upwind residual distribution schemes (RDS) for hyperbolic systems for application to compressible and incompressible flows. First, the design principles of RDS are explained for the simplest case of linear scalar advection. Extension to linear hyperbolic systems is described next. Then, their application to compressible and incompressible flows is discussed, and illustrative examples of applications are presented. Key Words: finite volume method, finite element method, compressible flows, incompressible flows

1.

Introduction

Multidimensional upwind residual distribution schemes, which were first introduced by P. L. Roe [ROE 87], have been developed on ideas borrowed from both the finite volume and finite element methods to become nowadays an attractive alternative to either one [PAI97]. The initial motivation for their development was a discontent about some drawbacks of the state-of-the-art finite volume solvers based on 1-D approximate Riemann solvers, namely • D by D first order upwinding is very diffusive, • ID Riemann solvers do not capture the real multidimensional flow physics, • higher order schemes use wide stencils. The starting point for the development of these schemes was a reinterpretation of ID finite volume schemes based on the concept of fluctuation [ROE 82]. Considering the continuous piecewise linear data representation classically used in finite element methods rather than the discontinuous cell-wise reconstructions used in finite

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volume methods, the flux difference between two nodes (F/+1 — F() is reinterpreted as the flux balance (else fluctuation or residual) over the interval (element) [i,i+ 1] (= fedF/dx dx), which is to be distributed to the vertices of the element. This interpretation can be carried over to several dimensions, provided the fluctuation is now the flux balance over a simplex (triangle/tetrahedron) to which is associated a continuous piecewise linear data representation. Because the elementary discretisation unit is an element rather than an edge or face for the finite volume method, the multidimensionality of the problem can genuinely be taken in to account: in 2D, the fluctuation can be distributed to the three vertices of the triangle rather than to the two neighbouring cells of an edge. Based on this idea, schemes were constructed, which combine a number of attractive features: • a much lower cross-diffusion than their finite volume counterparts, due to the genuinely multidimensional upwinding they incorporate, • a positivity property which ensures the satisfaction of a discrete maximum principle and consequently the absence of spurious oscillations, • (almost) second order accuracy on compact stencils. In addition, the compact discretisation stencil allows for the development of efficient implicit iterative solution strategies [ISS 96] and for an easy parallelisation [ISS 98, vdW99]. The paper starts with the presentation of upwind residual distribution schemes for linear hyperbolic scalar equations and systems, successively. This is followed by a discussion on their application to non-linear problems, specifically to the compressible and incompressible flow equations. Finally, the paper concludes with a few illustrative computational examples.

2.

Linear equations

2.1. 2.1.1.

Scalar advection Design principles

The residual distribution schemes (RDS) have been designed for an optimal discretization of the steady state convection equation

on PI finite element meshes, i.e. triangular (resp. tetrahedral) meshes with a piecewise linear solution representation. Evaluating the residual or fluctuation over an element

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Figure 1: Triangular element with scaled inward normals T, defined as the integral over this element of the differential operator, i.e.

since both /L and VM are constant over the element, one obtains §T — £,-&,-M,-, where ki = ^ A • nt is called the inflow parameter, ni being the scaled inward normal of the edge opposed to node i (see Fig. 1) and d the dimension of the problem. The method consists in distributing fractions of r to the vertices of the element. The resulting discrete equations therefore express that the nodal residuals Rf, sum of all contributions from neighbouring elements, vanish, i.e.

in which j8-r are the distribution coefficients. On each element T, these distribution coefficients must sum up to one for consistency and conservativity. The different schemes, corresponding to different ways of computing the distribution coefficients, have been designed to satisfy several properties making them optimal: UPWIND CHARACTER (W): No fraction of the element residual is sent to upstream nodes or /3^ = 0 when k{ < 0. POSITIVITY (^): The scheme does not create local extrema or, if we write the conT tribution to the element residual as 6? = Qj6 = y^J/ c -•• J« ,J, we impose c,,lj <— ' * 0 V; ^ i.

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Scheme N LDA SUPG PSI

Table 1 : Residual distribution schemes type <% K linear V W'Xy+-*7)-%*7K--K;) + linear %/ 1 , kj_ 1 h linear T T

(XA )A

(%>

t^S

V

-2nX||

max(0,)3/ v )/I I .max(0,j3/ v )

non-linear

V V

JS?^ V V V

LINEARITY PRESERVATION (jSf £*): The distribution coefficients jSf remain bounded as 0r —> 0 which is equivalent to zero cross-diffusion (second order accuracy) on a regular mesh. CONTINUITY CT^): The distribution coefficients must be continuous functions of both the advection and the solution-gradient directions. Several schemes satisfying some or all of these design criteria have been developed both in 2D and 3D. A complete summary is given in [PAI 97]. The schemes used in the present study are summarized in Table 1 (formulas valid both for 2D and 3D), where k+ = max(0,& ( ), &r = min(0,/c / ). Note that as a result of a generalization of Godunov's theorem, only non-linear schemes can satisfy both the & and %*& properties. 2.7.2.

Finite element interpretation

Residual distribution schemes can be viewed as generalised Petrov-Galerkin finite element schemes. Indeed, denoting wj the weighting function associated to node i of element T, the Petrov-Galerkin discretisation of the convection equation (1) can be written

Now, from the definition of r and from the linear approximation of u over T, assuming a constant advection vector A,, A- • VM is constant over T and equal to 0 r /£1 T , where Q.T is the surface of the triangle T (volume of the tetrahedron T in 3D). Hence, the previous equation becomes

For the Petrov-Galerkin discretisation to be identical to the residual distribution discretisation (3), the weighting functions co[ should satisfy the following relation :

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This equation is not sufficient to uniquely associate a weighting function &)? to a residual distribution coefficient /3(^ unless the functional form of the weighting function is prescribed and depends on a single parameter. Prescribing the weighting function to be the sum of the finite element shape function (jp- and a piecewise constant contribution similar to SUPG weighting functions [DEM 97], i. e. (of = cpt + aj yT, where aj is the upwind bias coefficient and JT is the piecewise constant function equal to 1 on element T and 0 elsewhere, the equivalence condition (6) yields

2.2.

Hyperbolic systems Considering now the non-commuting linear hyperbolic system

the formal extension of the residual distribution discretization (3) is

where the fluctuation Or is now a vector whose expression is Or — £.K(.U(. with K( being now an inflow matrix expressed as K- = ^ A^n (f, and where fl? is a distribution matrix. The design criteria for distribution matrices are the same as for distribution coefficients for the scalar advection equation, plus the additional requirement of invariance under a similarity transformation, i. e. UPWIND CHARACTER: j3(r = 0 when K,. < 0 (where K. < 0 means that all eigenvalues of K,, which are known to be real due to the hyperbolic nature of the system, are negative). r r r J condition becomes C,. < POSITIVITY: With Of' = /3 *«; O = Y. *^j C;ij; U ;j, the positivity 'j — 0 V; / i.

LINEARITY PRESERVATION: The distribution matrices ff remain bounded as

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INVARIANCE: Under the similarity transformation <9U = TdQ, the hyperbolic system (8) transforms into A^|^ = 0 where Ag£ = T~1A.£T. The discretization of the transformed equation being

we require that It results that if the system is diagonalisable, i. e. if the matrices A^ commute, then taking T as the diagonalising transformation, the transformed distribution matrix )3l- is the diagonal matrix of scalar distribution coefficients for each decoupled equation. Many of the schemes listed in Table 1 can be formally extended to systems. For example, the system-N scheme is defined by

whereas the system-LDA scheme is defined by

The non-linear PSI scheme on the other hand proves to be more difficult to generalize. Considering the scalar PSI scheme as a limited N-scheme, its formal generalization to systems is

However, the distribution matrix of the N-scheme f$7'N is not explicitly defined by Eqn. 11. For diagonalizable systems, the condition of invariance under similarity transformations suffices to define ($T'N and hence ^'PSI uniquely. For general noncommuting systems, the invariance condition is no longer sufficient. The system PSI scheme used in the numerical applications is based on one particular definition of [3T'N which satisfies this condition. Further details are given in [PAI 97]. The construction of non-linear positive and linearity preserving schemes for systems is still an ongoing research topic. New developments, based on the reinterpretation of the scalar PSI scheme as a blended N/LDA scheme are discussed in R. Abgral1's invited paper in this conference [ABG 99].

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Compressible flows

3.1.

Conservative linearisation

The residual distribution schemes presented in the previous section have been defined for linear problems. To apply them to non-linear problems such as the Euler equations, an essential ingredient is a conservative linearization, which consists in finding on each triangle T an average state U such that

where

so that, locally, the non-linear system of conservation laws d^^/dx^ = 0 is approximated by the linear system A^dU/dj^ = 0. Let's show that, with such a linearization, conservation is indeed satisfied. Summing up the nodal residuals over the domain £1, we have

and the contributions of internal edges cancel out (telescoping property). For the Euler equations, a conservative linearization is easily obtained as a multidimensional extension of Roe's linearization [DEC 93]. Indeed, assuming a linear variation of Roe's parameter vector Z [ROE 81] and since both the vector of conserved variables U and the fluxes F^ are homogeneous functions of degree 2 in the components of Z,

where Z = ^rf Svertices^i ^s ^e Pr°Per average state to ensure conservation. 3.2.

Transformations and preconditioning

The matrix distribution schemes presented above are invariant under a similarity transformation. It is nevertheless useful to apply a similarity transformation to the lin-

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earized hyperbolic system in order to achieve maximum decoupling for the following reasons. On one hand, this reduces the computational cost thanks to the considerable simplification of the distribution expressions and on the other hand, this allows to select different distribution schemes on the various decoupled systems/scalar equations. Specifically, with the similarity transformation dU = TdQ, the discrete equations (9) are rewritten

where O^ = Z,-Kg(.Q.. Partial decoupling can be achieved if the flux jacobians A^ have common eigenvectors. For the Euler equations, the flux jacobians A^ have one common eigenvector so that it is possible to decouple one scalar equation from the original system, leaving a coupled 3x3 system and one decoupled scalar equation in 2D. The transformation to symmetrizing variables defined as dQ = (-^,du,dp a2dp}1 accomplishes this task. The decoupled scalar equation is nothing else than the entropy advection equation, which is well-known to derive from the Euler equations. As shown in [PAI 97], additional decoupling may be achieved by preconditioning, namely the system of equations is rewritten as1

and the residual distribution method is applied to the preconditioned system between brackets. The optimal preconditioning was found to be the van Leer-Lee-Roe preconditioning [vLE 91], which allows to decouple one additional equation, namely the total enthalpy advection equation. 3.3.

Viscous terms

Viscous terms are discretised using the finite element interpretation of the schemes presented in section 2.1.2.. Specifically, the space discretisation of the divergence of the stress tensor V • t is

Now, since

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4. Incompressible flows For application to incompressible flows, two avenues have been explored. The first approach, followed by Waterson & Deconinck [WAT 97], is based on the fact that the incompressible Navier-Stokes equations are a symmetric advective-diffusive system [DEM 97], which can therefore be discretised using the system distribution schemes presented in section 2.2. using a collocated PI finite element representation for all variables. The collocated arrangement is made possible by the inherent pressure stabilisation effect of the system discretisation. The second approach, followed by Bogaerts et al. [BOG 98], is based on a stable finite element representation of the velocity and pressure fields (PlisoP2/Pl element) which therefore does not require any pressure stabilisation. It results that scalar residual distribution schemes can be used to discretise the momentum equations, i.e. using the finite element interpretation, the discretised equations are

where (oik is the residual distribution weighting function associated to the velocity node / and to the k\h component of the momentum equation, and i// is the shape function associated to the pressure node j. Note that, because of the upwind component of the weighting function co- k, upwinding is introduced in the discretisation of the pressure gradient. This turns out to be essential for the accuracy of the method [BOG 98]. 5.

Applications

Capabilities of upwind residual distribution schemes are now illustrated by a couple of computational examples. 5. /.

Inviscid transonic flow over the M6 wing

The ONERA M6 wing is a well documented testcase for three-dimensional flows from subsonic to transonic speeds [AGA 94]. The selected transonic case is Moo = 0.84, a = 3.06°. The grid consists of 316275 nodes and 1940182 tetrahedra. The far-field boundary is half a sphere with a radius of 12.5 root-chord lengths. The computation of this testcase with the present multidimensional upwind method employed the van Leer-Lee-Roe preconditioning, allowing a hybrid discretization. The decoupled

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advection equations for entropy and total enthalpy are discretized with the scalar PSIscheme, while the acoustic subsystem is discretized with a blended system LDA/Nscheme. To give an indication of how the multidimensional upwind method performs compared to standard finite volume methods, the same testcase has also been computed with the Jameson-Schrnidt-Turkel (1ST) artificial dissipation scheme, a matrix dissipation (MATD) scheme and the Roe upwind scheme with Venkatakrishnan's reconstruction. Plots for the pressure coefficient Cp and the relative total pressure loss, defined as 1 — pt/'ptoo, in the 80% span cross-section are shown in Fig. 2. From these figures it is clear that the multidimensional upwind method is much less dissipative in the leading edge region and that it has a better shock capturing. 5.2.

Inviscid transonic flow over a complete aircraft

To demonstrate the applicabilty of the present multidimensional upwind method to a complete aircraft configuration, the transonic flow over a generic model of the Falcon 2000 executive jet has been computed. The grid for a half model consists of 45387 nodes and 255944 tetrahedra. The selected testcase corresponding to the cruise condition is M^ — 0.84, a = 3.06°. The solutions computed with monotone first and second-order multidimensional upwind schemes are compared. For the first order computation the system N-scheme is applied directly to the full Euler equations. The second order computation employs the van Leer-Lee-Roe preconditioning, where the advection equations for entropy and total enthalpy are discretized with the scalar PSIscheme and the acoustic subsystem is discretized with the blended system LDA/Nscheme. Figures 3-4 compare the Mach number and entropy isolines for these two solutions. The better shock capturing and lower spurious entropy generation in the leading edge region of the wing and the engine pylon are clearly observed.

5.3.

Incompressible turbulent flow over a backward-facing step

We consider now the incompressible turbulent flow over a backward-facing step, experimentally studied by Kim [KIM 78]. Calling H the step height, the inlet channel width is 2H, so that the outlet channel width is 3H. The flow Reynolds number is 1.41 105 based on the inlet centreline mean velocity t/0 and the outlet channel width. The standard k — e turbulence model [JON 72] is used with wall function boundary conditions. The grid is a triangulated stretched Cartesian grid extending from 3H upstream of the step to 21H downstream. Along all solid walls, the computational domain boundary is set at a distance h — 0.025// from the wall. The grid contains 6247 PI nodes distributed along 107 vertical grid lines, where each vertical grid line contains 41 PI nodes in the inlet channel and 61 PI nodes in the outlet channel. The inlet boundary conditions for u, v and k are taken from the experiment and £

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Figure 2: M6 wing: Cp and total pressure loss distributions at 80% span, upwind residual distribution schemes versus finite volume schemes

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Figure 3: Falcon: Mach number contours

Figure 4: Falcon: Mach number contours is evaluated using a mixing length model. At the outlet, p is set to 0 and homogeneous Neumann (fully developed flow) boundary conditions are specified for all other variables. Along the wall function boundaries, a homogeneous Neumann condition is used for k and a Dirichlet condition is used for e (= c3/ 4 fc 3 / 2 / k h ). The wall shear stress is derived from the law of the wall. Fig. 5 show the flowfield pattern calculated using the LDA scheme. The reattachment length is 6.3H, a value closer to the experimental value of Lr/H — 7.0 than

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Figure 5: Turbulent flow over a backward-facing step: mean flow streamlines values obtained by other numerical computations using the same turbulence model, which fall in the range (5.7 < Lr/H < 6.0 [DEM 97]). The present method is also seen to capture a small secondary recirculation region at the foot of the step. Acknowledgements This paper summarises over 10 years of research at VKI, which was mainly carried out by a series of Ph.D. candidates: R. Struijs, H. Paillere, E. van der Weide, J.-C. Carette, E. Issman, N. Waterson, S. Bogaerts, K. Sermeus. The ONERA M6 wing and the generic Falcon computations were carried out within the collaborative IMT project IDeMAS funded by the European Commission. The reference computations of the ONERA M6 test case were performed by Daimler-Chrysler Aerospace. The grid for the generic Falcon configuration was provided by Dassault Aviation. References [ABG 99] R. Abgrall. Construction of genuinely multidimensional upwind schemes. 2nd International Symposium on Finite Volumes for Complex Applications, July 1999. [AGA 94] A Selection of Experimental Test Cases for the Validation of CFD codes. AGARD Advisory Report No 303, 1994. [BOG 98] S. Bogaerts, G. Degrez, and E. Razafmdrakoto. Upwind residual distribution schemes for incompressible flows. Fourth European Computational Fluid Dynamics Conference, Athens, Sep. 1998. [DEM 97] T. De Mulder. Stabilized finite element methods for turbulent incompressible singlephase and dispersed two-phase flows. PhD thesis, K. U. Leuven, Leuven, Belgium, 1997. [DEC 93] H. Deconinck, P. L. Roe, and R. Struijs. A multidimensional generalization of Roe's

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Finite volumes for complex applications flux difference splitter for the Euler equations. Computers and Fluids, 22:215-222, 1993.

[ISS 98]

E. Issman and G. Degrez. Non-overlapping preconditioners for a parallel implicit Navier-Stokes solver. Future Generation Computer Systems, 13:303-313, 1997/1998.

[ISS 96]

E. Issman, G. Degrez, and H. Deconinck. Implicit upwind residual-distribution Euler and Navier-Stokes solver on unstructured meshes. AIAA Journal, 34(10):20212029,1996.

[JON 72] W. Jones and B. Launder. The Prediction of Laminarization with a Two-Equation Model of Turbulence. Int. Journal of Heat and Mass Transfer, 15:301-304,1972. [KIM 78] J. S. Kim. Investigation of separation and reattachment of a turbulent shear layer: Flow over a backward-facing step. PhD thesis, Stanford University, Stanford, Ca, 1978. [PAI 97]

H. Paillere, H. Deconinck, and E. van der Weide. Upwind residual distribution methods for compressible flow: an alternative to finite volume and finite element methods. VKI LS 1997-02,1997.

[ROE 81] P. L. Roe. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of computational physics, 43:357-352,1981. [ROE 82] P. L. Roe. Fluctuations and signals, a framerwork for numerical evolution problems. In K. W. Morton & M. J. Baines, editor, Numerical Methods for Fluid Dynamics, pages 219-257. Academic Press, 1982. [ROE 87] P. L. Roe. Linear advection schemes on triangular meshes. CoA Report 8720, Cranfield Inst. of Tech., 1987. [vdW 99] E. van der Weide, H. Deconinck, E. Issman, and G. Degrez. A parallel, implicit, multidimensional upwind residual distribution method for the Navier-Stokes equations on unstructured grids. Computational Mechanics, 23(2): 199-208, 1999. [vLE 91]

B. van Leer, W. T. Lee, and P. L. Roe. Characteristic time stepping or local preconditioning of the Euler equations. AIAA Paper 91-1552-CP.

[WAT 97] N. P. Waterson and H. Deconinck. A fully-implicit multidimensional upwind approach for the incompressible Navier-Stokes equations. In C. Taylor, editor, Numerical methods in laminar and turbulent flows, volume X. Pineridge Press, 1997.

Overcoming mass losses in Level Set-based interface tracking schemes

Th. Schneider and R. Klein Konrad-Zuse-Zentrum fur Informationstechnik Berlin, Germany FB Mathematik & Informatik, Freie Universitdt Berlin, Germany

ABSTRACT An extended level set method is presented for tracking material interfaces in incompressible two-phase flow that ensures conservation of mass. Inconsistencies, between mass transport and interface (level set) motion due to truncation errors are reconciled by two correction steps. Step 1 involves a redistribution of mass from cells that are not intersected by the tracked front but carry unphysical intermediate densities. Step 2 controls deviations between level set and density based partial volume fractions by introducing a small correction velocity in the level set transport equation. Key Words: level set, VOF, interface tracking, incompressible two-phase flow

1. Introduction Among the approaches to compute incompressible flows with material interfaces such as level set methods, volume of fluid methods (VOF) and interface tracking schemes, level set methods currently attract considerable attention. Since level set methods only require solution of a scalar hyperbolic transport equation, they are simple from an algorithmic point of view, their implementation is straight forward and the computational effort to solve the scalar transport equation is negligible in comparison to the effort an incompressible method requires. Furthermore, level set methods naturally support interface distortion as well as topological changes of the interface. The major drawback of level set methods is that they do not ensure conservation of mass by construction. Volume of fluid methods consist of two parts : reconstruction of the interface and advection of the volume fraction. The first part implies a considerable algorithmic effort since the only available information concerning the position of

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the front is hidden in the distribution of the volume fractions. By construction volume of fluid methods conserve mass; in fact for piecewise constant densities, the volume fraction transport equation is equivalent to mass conservation. This is not true in more general cases, though.

2. Level set methods

In the framework of tracking material interfaces in incompressible flows the underlying idea of the level set approach is to identify the phases by the sign of a scalar field G. In other words : areas where G > 0 are occupied by phase I, whereas areas where G < 0 are occupied by phase II. Therefore, the location and the topology of the interface are described by the zero level set of G. Since the interface is advected along particle paths the evolution equation for G is:

Sussman et al. [Sus 94] were the first to use a level set formulation to represent interfaces in incompressible flow. They performed computations of incompressible two-phase flows with density ratios up to 1000:1; they observed considerable mass losses. In order to reduce mass losses they reinitialized G after each time step. Their reinitialization ensures that G remains a distance function, i.e. ||VG||2 = 1. This is achieved by performing a pseudo time integration of

until a steady state is reached. The crucial information that is needed from the distribution of G is the zero level set that represents the interface. In areas where G > 0 or G < 0 the scalar field G is, up to its sign, physically meaningless. This implies that any change may be applied to G in those areas as long as the sign of G remains unchanged. Adalsteinsson et al. [Ada 99] proposed a numerical method to solve the transport equation (1) of G by constructing appropriate extension velocities such that G remains a distance function. Chang et al. [Cha 96] used another reinitialization that ensures conservation of area and by that conservation of mass. They reinitialize G by integrating

until a steady state is reached. Herein e is a small positive constant, K, is the local curvature of the front. Even though this ansatz overcomes the key

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problem of loss of volume when the advecting velocity field is divergence free, it is insufficient for our purposes: The ultimate goal of the present efforts is to construct a flow solver for general zero Mach number variable density flows with phase interfaces. Thus we must combine the level set description with a discretization of mass conservation. This is not achieved in [Cha 96]. 2. Volume Of Fluid Methods

A detailed description of numerous variants of VOF methods can be found in Rider et al. [Rid 95], [Rid 95/2] and [Rid 98]. All VOF methods have in common that they consist of two parts: 1. Reconstruction of the interface : Since no direct information on the topology (normal vector n, curvature K) of the interface is available at each time step the interface needs to be reconstructed based upon the distribution of the volume fractions / in order to allow the advance in time. 2. Advection of the volume fraction /: The volume fraction / is advected according to the following equation:

The flux F is constructed using the information of the interface motion that is obtained in the first part; V is an arbitrary control volume, e.g. grid cell. The numerical method that is used to solve the advection must ensure that / remains bounded : 0 < / < 1. Therefore, if this latter condition is violated mass will be redistributed to mixed cells in the neighbourhood that can absorb it.

3. A coupled Level-Set/Volume-Of-Fluid Algorithm

Bourlioux [Bou 95] presented a coupled level set/volume of fluid method. Her basic idea was to solve an advection equation for the volume fraction / in a VOF manner and the scalar G at the same time. The geometric information of the interface needed to advect / is obtained from a scalar field G; therefore, a reconstruction of the interface from the VOF step function data is not required. During the reinitialization of G she applied an additional correction to

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G in order to minimize the difference of the volume fraction / and the volume fraction fa that results from the integration of the distribution of G within a cell according to

H(G) denotes the Heavyside function.

4. A coupled Level Set / mass conservation Scheme Our proposed method is different from Bourlioux's work in many respects even though the basic idea remains the same. The advection equation of the volume fraction / can be written as follows (the volume fraction f is associated with the phase to which the positive sign of G is assigned) :

H(G) denotes the Heavyside function. If the control volume V is a grid cell with HI cell interfaces a first order discretization of the above advection equation of the volume fraction / is :

n+2

Where /^

is a second order approximation for :

Equation (7) does not automatically ensure that fn+l remains bounded 0 < fn+l < I which results from truncation errors. Therefore, a correction A/ n+1 needs to be added to /n+1. In order to conserve mass A/n+1 is introduced as correction of the fluxes in equation (7) according to :

The equation above is a discrete Poisson equation for the scalar that establishes an implicit coupling between all mixed cells along the interface. Using

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this correction (/n+1 - fn+1 + A/ n+1 ) equation (7) changes to:

A second correction is applied to the scalar field G in order to control the difference of the partial volume fraction fn+l and f£+l- The idea is to locally move the front normal to itself in order to minimize this error. To determine a local correction velocity s* a piecewise linear reprensentation of the front is assumed. The local correction velocity is set to s* — (h — ho)/At (see sketch below): This local correction velocity s* is used to perform one time step of the

Figure 1: Piecewise linear representation of the front within a mixed cell following equation:

5. Results 5.1 Advection of a circle

To test the convergence rate of our proposed extended level set method we have chosen a circular density jump which is passively advected within a

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parallel flow. The unit square domain is resolved by 32 x 32, 64 x 64 and 128 x 128 grid cells. Periodic boundaries are imposed. The LI error GI of the density is measured taking the difference of a coarse grid and fine grid solution according to :

rih is the number of grid cells of the fine grid in each direction, p2/l the density of the coarse solution and ph the density of the fine grid solution. The rate of convergence p1 based upon two L\ error approximations eij 64,32 and ei) 128,64 is given by :

The initial data are : radius 0.15, centered at (x0,yo) — (0.5,0.5), density ratio 1000:1, UOQ = 0.25, t>oo = 0.25. When integrating from t=0.0 to t=0.5 a convergence rate of 1.8 was achieved.

5.2 Rising bubble The governing equations for the considered variable density zero Mach number regime are (using the pressure decomposition p = PQ + M2£/2\ where -J^PQ = 0, since the flow is considered as incompressible):

H denotes a smoothed Heavyside function. The numerical method for zero Mach Number variable density flow is described in [Sch 98]. A rectangular domain with a height to width ratio of 2 was chosen consisting of 128 x 256 grid cells. At the boundaries slip conditions were set. This problem is characterized

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by the density ratio of liquid and gas C\, the diameter to width ratio C^, the viscosity ratio C3, the Reynolds number Re, the Froude number Fr and the Weber number W. The characteristic number and their definitions are listed below:

Ci C2 C3

Pi / Pg D/LX Vl/Vg

714 0.25 6667

Re Fr W

pi Uref D / pi

D2/a

9.7 0.78 7.6

Surface tension is implemented in the same way as explained in [Sus 94]. The initial shape of the interface is circular. The interface (thick line), streamlines (thin lines) and velocity arrows are plotted for times £=0.0, 1.0, 2.0, 3.0 below. In figure (3) the relative mass error (mass according to the distribution of G) is plotted versus time.

Figure 2: Interface (thick line), streamlines (thin lines) and velocity arrows at times £=0.0, 1.0, 2.0, 3.0 6. Concluding Remarks

The present scheme for fluid interface tracking methods avoids mass losses by incorporating a finite volume mass conservation equation. The inconsistencies between level set transport and mass conservation are eliminated by controlling volume fraction deviations through a suitable "penalty" - method

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Figure 3: Relative mass error resulting from the distribution of G versus time applied to the level set transport. Unphysical intermediate densities in nonmixed cells which can arise due to the capturing features of the mass conservation discretization are removed by a mass redistribution step. In our present approach the thickness of the intercase is zero. Artifical smearing of densities is neither necessary nor does it occur during a computation. Acknowlegdments

This work is supported by the Deutsche Forschungsgmeinschaft through project KL-611/5 - 1,2, in the framework of the CNRS-DFG Programme "Numerische Stromungssimulation" . Helpful discussions with Dr. Anne Boulioux (Montreal) and Dr. Raz Kupferman (Jerusalem) are appreciated.

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Bibliography

[Ada 99]

Adalsteinsson, D. und Sethian, J. A., The Fast Construction of Extension Velocities in Level Set Methods, Journal of Computational Physics, 148, 2-22, 1999

[Bou 95]

Bourlioux, A., A Coupled Level-Set Volume-Of-Fluid Algorithm for Tracking Material Interfaces, 3rd Annual Conference of the CFD Society of Canada, June 25-27, 1995, Banff, Alberta, Canada

[Cha 96]

Chang, Y. C., Hou, T. Y., Merriman, B. und Osher, S., A level Set Formulation of Eulerian Interface Capturing Methods fo Incompressible Fluid Flows, Journal of Computational Physics, 124, 449-464, 1996

[Puc 97]

Puckett, E. G., Almgren, A. S., Bell, J. B., Marcus, D. L. and Rider, W. J.,A High-Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows, Journal of Computational Physics, 130, 269-282, 1997

[Rid 95]

Rider, W. J., Kothe, D. B., Mosso, S. J., Cerutti, J.H. and Hochstein, J.I. Accurate Solution Algorithms for Incompressible Flows, 33rd Aerospace Sciences Meeting and Exhibit, January 9-12, 1995, Reno, NV

[Rid 95/2]

Rider and W. J., Kothe, Stretching and Tearing Interface Tracking Methods, 12th AIAA CFD Conference, June 20, 1995, San Diego. Paper number AIAA-95-1717

[Rid 98]

Rider, W. J. and Kothe, D. B., Reconstructing Volume Tracking, Journal of Computational Physics, 141, 112-152, 1998

[Sch 98]

Schneider, T, Botta, N, Geratz, K. J. and Klein, R., Extension of finite volume compressible flow solvers to multidimensional, variable density zero Mach number flow, submitted to the Journal of Computational Physics, preprint available at http://www.zib.de/thomas.Schneider/pub.html

[Set 96]

Sethian, J. A., Level Set Methods, Cambridge University Press, 1996

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[Sus 94]

Sussman, M., Smereka, P. und Osher, S., A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow, Journal of Computational Physics, 114, 146-159, 1994

[Sus 99]

Sussman, M., Almgren, A. S., Bell, J., Colella, P., Howell, L. H. und Welcome, M., An Adaptive Level Set Approach for Incompressible Two-Phase Flows, Journal of Computational Physics, 148, 81-124, 1999

Coupling mixed and finite volume discretizations of convection-diffusion-reaction equations on nonmatching grids

Raytcho D. Lazarov, Joseph E. Pasciak, and Panayot S. Vassilevski Department of Mathematics, Texas A&M University, College Station, TX 77843 and Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.A.

ABSTRACT In this paper, we consider approximation of a second order convectiondiffusion problem by coupled mixed and finite volume methods. Namely, the domain is partitioned into two subdomains, and in one of them we apply the mixed finite element method while on the other subdomain we use the finite volume element approximation. We prove the stability of this discretization and derive an error estimate. Key Words: combined mixed and finite volume methods, non-matching grids.

1. Introduction Coupling different numerical methods applied to different parts of the domain of interest is becoming an important tool in numerical analysis, scientific computing, and engineering simulations. In the coupling process several important mathematical issues arise that have to be addressed. First problem is to find what natural and stable mathematical formulation will lead to a good computational scheme. In the case of different methods used in different parts of the domain this means to find a stable way of gluing together the solutions in the subdomains. Secondly, we have to find an approximation of the mathematical formulation which is stable, convergent, and accurate. And finally, we have to construct and study efficient solution methods for the resulting algebraic problem.

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We shall consider the following homogeneous Dirichlet boundary value problem for the convection-diffusion-reaction equation:

where C — £Q + C, Cop = — V • aVp is the diffusion operator, and Cp = V • (pb] + CQP is the convection-reaction operator. Here Q is a bounded polygon in 7£d, d = 2,3 with a boundary <9fi, a = a(x] = {a^j(x)} is a d x d symmetric and uniformly in fJ positive definite matrix, and / = f(x) is a known function in L 2 (fi). Also b = b(x) = (bi, • • •, bj) is a given vector field and c0 = CQ(X) is a given function. We assume that b_(x] and CQ(X) are uniformly bounded in fi and satisfy the condition

This in turn guarantees the coercivity of the operator C in L 2 (f2) and the existence and uniqueness of its solution in the Sobolev space H01(^,}. This problem is a prototype of mathematical models in heat and mass transfer, diffusion-reaction processes, flow and transport in porous media, etc. In this paper we propose and study numerical methods for this problem when in different parts of the domain different discretizations on independent meshes are used. Namely, we consider mixed finite element approximation in one part of the domain and finite volume element method in the rest of the domain. It is important to note that coupling mixed finite element and finite volume or Galerkin finite element approximations does not require any auxiliary (mortar) space on the interface of the subdomains. This is due to the fact that the Dirichlet boundary conditions are natural for the mixed formulation, while the Neumann boundary conditions are natural for the standard weak formulation of a second order elliptic problem. In the recent years there has been growing interest in the finite volume method (called also control-volume method or box-schemes). This interest is mostly due to the requirement of many applications of having locally conservative discretizations. This is a discrete variant of the property of the continuous model which expresses conservation of certain quantity (mass, heat, momentum, etc) over each infinitesimal volume. The finite volume method has been combined with the technique of the finite element method in a new development which is capable of producing accurate approximations on general simplicial and quadrilateral grids (see, e.g. [4, 5, 6, 7, 8, 13]). For a collection of theoretical results and various applications we refer [2]. The main advantages of the finite volume method are compactness of the discretization stencil, good accuracy, and discrete local conservation, which for many applications is a very

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desirable feature of the approximation. Also, this method has well developed approximation schemes for convection and convection-dominated problems. The structure of the paper is the follow. In Section 2 we introduce all necessary notations and the weak form of the problem (1) in a two domains setting. In the subdomain where the mixed formulation is used we apply the more general concept of the discontinuous Galerkin method for second order equations in mixed form. In the case of convection-diffusion problems the pressure should be smoother than just L2 so we use the space H]oc- Further, we study the stability and derive an a priori estimate for the solution. Section 3 is the central part of the paper. Here we introduce and study the coupling of the mixed finite element and the finite volume element method. Further, in Subsection 3.2 we discuss the coupled mixed and finite volume approximation of convection-diffusion-reaction equations. Finally, in Subsection 3.3 we prove the unconditional stability of the discrete scheme and derive an estimate for the error. 2. Variational formulation In this section we first introduce all necessary notations for splitting the domain of the problem (1) in two subdomains 17 = $7i U 1^2 and using two different formulations in each subdomain. The weak mixed formulation in QI is derived when the pressure p is in the space jf//oc(17i). In ^2 we use pressure space Hl(ft,2). We prove that the coupled mixed/Galerkin formulation is stable and derive an a priori estimate which is the prototype of estimates for the approximations schemes established further in the paper. 2.1. Two-sub domain coupled formulation We partition ill into two subdomains with an interface boundary F, i.e. 17 = 17i U F U 172 (see Figure 1) and use the standard notations for Sobolev spaces of functions defined on 17i and 172: H(div, 17i), L2(17j), i =• 1,2 and #o(172, <9172 \ F). Here the last space denotes the functions defined on 172 having generalized derivatives in L 2 ((7 2 ) and vanishing on 5172 \ F. The inner products in these spaces are denoted correspondingly by

and

by II

, which in turn define norms denoted

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Figure 1: Domain partitioning: £7 = HI U F U Q2-

Whenever possible we skip the subscript in L2-norms. The dual space of HQ (fJ 2 , 90,2 \ F) is denoted by H~l(£l-2), the space of the traces of the normal component of the vector-functions in H(div, f ^ i ) is denoted by ff~ 1 / / 2 (F), and the space of the traces of functions in #o(Q 2 , <9n2 \ F) is denoted by 1 /2 HOQ (F). The trace spaces are equipped with the standard Sobolev norms. Finally, we shall use the notation < •, • >r for the duality pairing between HQQ (F) and H~l/2(T). Further, we denote by n; the unit vector normal to dtli for i = 1,2 and pointing outward to the domain. Finally, we split the interface boundary F = F_ U F+ where F_ — {x € F : b(x) -HI < 0} and F+ = {x £ F : b(x] -HI > 0}. Note, that this splitting is with respect to the vector m. An illustration of these notations in 2-D is given on Figure 1. In QI we use a mixed setting of the problem (1). That is, we introduce the new (vector) variable u = — aVp. To distinguish the solutions in the subdomains we denote by pi = p\Qi and p2 = Pln 2 - The composite model will impose different smoothness requirements on the components p1 and p2More specifically, we will require that u € #(div, fii), p\ 6 L 2 (fii), and P2 G .H0(f2 2 , dft \ F). Note that p2 is required to vanish on <9£)2 \ F. Testing the equation a-1u + Vpi = 0 by a function v e H(div, 17), using integration by parts, the zero boundary conditions for p\ on d£l\ \ F, and the fact the trace of p\ on F is the same for the trace of p2 on F, one ends up with the equation,

Further, in order to describe the weak form of the equation

we need to allow discontinuous functions pi from the space

The functions in Hfoc(tli} have traces from both sides of the interfaces of the subdomains K. Namely, for a given function pi E Hioc(fli} we denote these traces by p° and p\, where "o" stands for the outward (with respect to K} trace and respectively, "i" stands for the interior trace. Next, we give the weak form of the above equation. We borrow this formulation from the discontinuous Galerkin methods (see, e.g. [10], pp. 189-196) by testing the equation by a function w\ G Hfoc(Qi). We note, that this setting is quite similar to the mixed finite element method for convection-dominated convection-diffusion-reaction equations (see, e.g. [12]). Since the functions from HIOC(£II) are piece-wise smooth with respect to the partition JC we shall integrate over each K 6 /C and then sum the results. Following [10] we find first the contributions of the advection-reaction operator Cp\ by introducing the bilinear form CK(PI, MI) for any subdomain K € JC:

Here n is the outer unit normal vector to dK. Next, we integrate by parts in each subdomain K and sum over all K € 1C. Thus, for pi, w\ G Hloc(£l\) we get:

Note that this bilinear from is well defined for both continuous and discontinuous functions with respect to the partition /C. From this expression we see that if the subdomain K has a side/face on F_ then the trace p° should be replaced by its counterpart from f7 2 , namely by Pi(x}. Also on F_ we have w{ — w\ and on <9£7i_ \ F_ we take p° = 0. Further, for a given function t(x) we denote by £_ = min(0, t) and t+ = max(0, t). Thus, we get the following weak form of the second equation valid for all w\ 6 ///oc(£)i):

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where

and

Finally, testing the equation (1) by a function W2 € £[$(0,2] d^t-2 \F), using integration by parts, the zero boundary condition for w^ on 5fJ2 \ F, and the fact that u • ni = — aVpi • ni = aVp2 • 112 on F, one arrives at,

for all There are various ways one can take into account the influence of the problem in the domain fii on the problem in D2. One of the possibilities, which we shall use further, is to try to make a formulation, which is stable for small diffusion coefficient (or even for vanishing diffusion). In this case it is very important to formulate correctly the boundary conditions. Namely, at the "inflow" part of the interior boundary the solution should be specified from the "outside" data. Taking into account that F+ is the "inflow" part of F for the subdomain 02, we add fr Piw? b-112 ds and subtract its equal Jr p^w? b-n.2 ds since on F we have pi = pi. Thus, we get the following form of the last equation:

for all

where

Thus, the coupled system for the three unknowns u e H(div, fix), p\ 6 H\oc($l\) and p2 G #o(02; <902 \ F) consists of the equations (3), (5), and (9)

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summarized as:

for all v E H(div, fti), wi e Hloc($li], and w2 G #0(^2; 5H 2 \r), respectively. The bilinear forms %•(-, •) are defined by (6), (7), (10), and (11), respectively. 2.2. Well-posed-ness of the composite problem Here we verify the existence and uniqueness of the solution of problem (12) and its stability in an appropriate norm. For this we shall need some additional notations. Let £, = {e} be the set of edges/faces of the subdomain QI from /C and £0 the set of interior for fii edges/faces. Recall, that HI and n2 are the outward unit normal vectors to HI and J7 2 , respectively. For any edge e G £Q denote by ne a fixed unit vector normal to e and let K+ and K~ be the two adjacent to e subdomains from the partition /C. For edges/faces that are on dfli we shall always assume that ne = ni. Further, denote by [vi] and v\ the jump and the average of the discontinuity of v\, respectively, along any edge e. More precisely, this is the difference and the arithmetic mean of the traces VI\K+ and VI\K- taken from both sides of e:

Further, we use the following natural norm for

All terms in the expression on the right are nonnegative and this defines a norm on the space Hloc($li] x ./^(f^; <9fJ2 \ H- Note, that under certain conditions on the vector field b this is a norm even if 70 =0.

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The stability of the composed problem (12) is based on the following theorem: Theorem 1 The solution of the problem (12) satisfies the a priori estimate:

where the *-norm is defined by (13). The proof of the above theorem is based on the following lemmas. Lemma 1 The bilinear form (6) defined for v\, w\ G #/ oc (fJi) can be transformed to the following form:

Furthermore,

Lemma 2 For all

3. Coupling mixed and finite volume approximations of the convectiondiffusion equation Our approximation strategy is based on the finite volume method in the framework studied by Cai [4], Cai, Mandel, and McCormick [6] and also by Bank and Rose [1].

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3.1. An outline of the finite volume element method We first outline a finite volume discretization method for the case of pure diffusion problem posed on fi2,

Here / £ L2(^2) is given and TJN, for the time being is assumed given in the space L2(T). The finite volume method under consideration uses two different finite dimensional spaces: a solution space W2 and the test space W% • The Wi is the standard conforming space of piecewise linear functions over the triangulation Ti = {T} of f)2 into triangles in 2-D and tetrahedra in 3-D (we call them simplices). To introduce the test space W% we need a dual partition V2 of the domain into the finite (control) volumes V. Let M denote the set of all vertices (nodes) of the triangles/tetrahedra from ?2 and let A/o be a subset of those vertices that are not on the Dirichlet part of the boundary <9H2 \ F. In each simplex T £ ?2 one selects an interior node XT- Next, in 2-D one links XT with the midpoints of the sides of the triangle. In this way the triangle is split into three quadrilaterals. In 2-D, one can select XT to be the orthocenter of the finite element T and then the edges of the volume V(x) will be the perpendicular bisectors of the finite element edges (see the right Figure 2). With each vertex x G A/" of a simplex from ?2, we associate a volume V = V(x) that consists of all quadrilateral/polyhedra having x as a vertex (see Figure 2 for finite volumes in 2-D). The splitting of £1% into finite volumes V forms the partition V2 (see, Figure 3). Consider now the test space W£ spanned by the characteristic functions of the volumes V G V2 and that vanish at the nodes M \ A/o on the boundary d^l-2 \ F. If one defines the piecewise constant interpolant I£ with respect to the volumes V G V2, then the space W% is actually equal to I^W-i because they have the same degrees of freedom (associated with the vertices x € A/")-

The L 2 (ft 2 ) and Hl(ft,2) norms in W2 are defined in a standard way. We shall need also discrete variants of these norms for functions in W2*. First, we define the interpolation operator //> : W£ i—> Wh by the following natural rule: Ih^2 is ^ne piece-wise linear interpolant of v2 over each finite element T G ?2Then we define H^lli./i = 11-^2 111,n 2 - This norm is essentially formed by the squared differences of the values of v% at the vertices of each finite element.

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Figure 2: A finite volume V associated with a vertex from the primal triangulation. Left: the vertices of V interior to the triangles T are arbitrary, whereas those on the edges of T are midpoints.

Figure 3: In HI we use the lowest order Raviart-Thomas spaces over the finite elements T; in ^2 we use a solution space W% of continuous functions that are linear over the finite elements T and a test space W£ of pice-wise constant functions over the volumes V.

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Next, we define the following, in general nonsymmetric, bilinear form on W2 x W%:

Then the finite volume approximation of (15) is: find v% G W^ which satisfies the following identity for all w\ G W2*;

We note that the integrals over dV for V = V(x) a volume corresponding to a Neumann node x (i.e., if a: € F), contain only the interior (to f^) part of dV. We assume that the triangulation TI is aligned with the possible jumps of the coefficient matrix a(x), i.e. over each finite element T G ?2 the matrix a(x) has smooth elements. Therefore, there is a constant CQ > 0 such that for all

where These inequalities of two d x d matrices with real elements are understood in the sense of inequalities for the corresponding bilinear forms, i.e. a > a(x), iff £Ta£ > £Ta(:r)£, V £ G 7ld. Also, the above equality of the matrices 0(2;) and d(x) is understood in element-by-element sense, i.e. the elements of a(x] are the mean values over T of the corresponding elements of a(x). Obviously, in case of piece-wise constant coefficients a(x) = a(x] and CQ = 0. The well posed-ness of the finite volume element approximation follows from the weak coercivity of the bilinear from 02,71(^2,^2) f°r sufficiently fine partitions ?2. We have: Lemma 3 Let the partition Ti be so fine that h < I/Co, where the constant CQ is determined in (17). Then the following inequality holds true

with a constant C independent of h.

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In fii we use the lowest order Raviart-Thomas spaces. Thus, V as a subspace of H(div; fii), and W\ is the space of piece-wise constant functions with respect to the partition 71 and therefore a subspace of Hioc(Q,i) for /C = 71. Thus, the advection term Cp\ will be discretized using the pressure space W\ of piecewise constant functions on the triangulation 71. For the discretization in £1-2 we will use the space Wz C #o(fJ 2 , <9^2 \ F) of continuous piecewise linear functions on 72• Further, in the finite volume setting we use as a test space W% of piecewise constant functions on V2. Thus, applying equation (4) consecutively for (ui, w\) € W\ x W2, and (i> 2 ,u> 2 ) € W2 x W£, respectively, we get the mixed finite element and finite volume approximations, respectively, of the bilinear form corresponding to the first order term. However, like in the standard Galerkin finite element method this approximation of the operator C will lead to central differencing, which in turn will lead to a conditionally stable (only for sufficiently small step-size h) scheme. In order to derive a unconditionally stable scheme we shall use upwind approximation in 02. 3.2. Derivation of the coupled method

Since both v\ and wi are discontinuous piece-wise constant functions with respect to the triangulation 7i the formula (4) is applied in a straightforward manner for JC — T\ so we get the following approximations a^ and a^2 of the forms an and 012, respectively:

and

Now we find the contributions of the the operator C from 02 and we define the approximations of the bilinear forms a2i and a 22 . We shall simply rewrite

Since the functions in W^ are continuous then C(i; 2 ,w 2 ) is well defined for all V2 € W-z and u>2 G W£. Taking into account that the functions in W£ are

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Figure 4: The shaded area is the volume V centered at the vertex xl; the doted boundary of V denotes the "inflow" boundary, while the solid one is the "outflow" part; on the various pieces of the boundary dV we have the following approximation of the convection term: on (a, 6), (m,c), ( / , g ] and (g, a): v2(x) - v 2 ( x l ) ; on (6,m) : v2(x] - v2(x°] = v2(C); on (c,d) : v 2 ( x ) = v2(x°) - v2(D}] on (d,e) : v2(x) = v2(x°) - v 2 ( E ) ; on (e,f) : v2(x] = v2(x°)=v2(F). constant over each finite volume V 6 V2 then the contributions from each finite volume V G V2 are:

Since v2 is continuous then obviously, we have v2 = v2 = v2(x). On the boundary F+ the values v2 are not defined (this is the inflow boundary for Q2) and we shall take them from the corresponding counterpart in fii, i.e. as vi(x). Thus, we split the integrals over dV into two parts and get

Unfortunately, the exact calculation of the first integral in (21) will lead to central differences and therefore to a scheme which is stable only for sufficiently small step-size h. The limitation of the step-size h will depend on the magnitude of the convection coefficient 6 relative to the diffusion coefficient (matrix) a. For problems with dominating convection this will lead to prohibitively small h. In order to avoid this conditional stability we introduce an up-wind approximation of the integrals. This approximation is done in the following way. We denote by

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V(xl) a finite volume centered at the vertex xl and by V(x°) any of neighboring volumes centered at the vertices x°. The integral over dV \F_ is split into subintegrals over the boundaries of V 7 = V(xi] D V(x°) n T with its neighboring finite (control) volumes and contained in the finite element T. We assume that over each 7 the function b(x) • n does not change sign (i.e. is either nonnegative or negative). Then on 7 we use upwind approximation of the following form (for a 2-D illustration, see Figure 4):

Note, that in the finite volume V(xl) we have I^V2 (x) = v^x1}. Similar equalities are valid for the neighboring volumes V(x°] as well. Thus, roughly speaking the function v?,(x) has been replaced by its interpolant in the space of discontinuous functions W£ and then taken the appropriate (up-wind or in the opposit direction of the vector-field &(#)) values at the finite volume interfaces. A particular finite volume in 2-D is shown on Figure 4. Summing for all V € V^ we finally get the following form by taking also into account the diffusion term (16):

for all

and the form

The coupled mixed finite element/finite volume approximation of the composite problem (12) reads as: find u/! e V, pi^ 6 W\, and p2,h € W%, such that

for all

, and

respectively.

3.3. Stability of the coupled scheme and error estimate

An important feature of the described above discretizations is that the corresponding operator is coercive in an appropriate norm and the method is stable.

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For proving the stability we shall follow the same argument as in the case of original setting. Let as before £ = {e} be the set of edges/faces of the elements from 7i and let £0 be the set of interior edges/faces. Similarly, Q — {7} is the set of edges/faces, with each 7 being the boundary of two adjacent volumes Vi G V<2 and V2 G V^ contained in a finite element T, i.e., 7 = V\ n V-2 n T. This splitting can be also used in the computational procedure, since it will lead to element-wise contributions of the convection term to the stiffness matrix. Note, that all edges/faces 7 are in the interior of f^- For the coercivity of the coupled problem we need the following discrete variant of the norm (13):

Here a is a piece-wise constant matrix with respect to the partition ?2 defined

by (17). Theorem 2 Let h < I/Co, where C0 is defined by (17). Then the solution of the problem (12) satisfies the a priori estimate:

where the (*,h)-norm is defined by (25) with respect to the partitions 71 and T 2

Proof: As in the continuous case, by testing (24) with and w-2 — —Ifrp2,h we get the equation:

Further, the estimate (14) is a consequence of the simplified form (28) of o^ and (29) of a^, which are established in the lemmas below. Lemma 4 For any edge/face e denote by ne a fixed unit vector normal to e and let T+ and T~ be the two adjacent elements to e. Similarly, for any

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edge/face 7 £ Q denote by n7 a unit vector normal to 7 pointing to one of the neighboring volumes Vj~ and V~. Also, [v\\ denotes the jump of a function across an underlined boundary (here we have either e or j) and w\ denotes the arithmetic mean of the jump (introduced in Section ). Then,

Similarly, for all v% € W-z, w% £ W% the following identity is valid (to the expressions we have used the notation v^ = l^i)'-

simplify

Proof: The proof of (28) and (29) essentially repeats the arguments of Lemma 1. There is a small difference in the proof of (29) where the integrals over each 7 6 G have been computed by using up-wind approximation. Lemma 5 The following identity is valid for all

Similarly, for all

W<2 (here in order to simplify we use the notation

Finally, we have the following error estimate:

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Theorem 3 Assume that the solution p(x] of the problem (1) is H2 -regular in 0. Then the solution (u.h,P\,h,Pi,h) of the coupled mixed discontinuous finite element and finite volume methods (24) converges to the solution (u,pi,p2) of the composite problem (12) and the following error estimate holds true:

The constant C does not depend on h but may depend on the ratios

and

4. Bibliography [1]

R.E. BANK AND D.J. ROSE, Some error estimates for the box method, SIAM J. Numer. Anal. 24(1987), 777—787.

[2]

F. BENKHALDOUN AND R. VILSMEIER (Eos), Proc. First Intern. Symposium on Finite Volumes for Complex Applications, July 15 18, 1996, Rouen, France, Hermes, 1996.

[3]

F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991.

[4]

Z. CAI, On the finite volume element method, Numer. Math., 58 (1991) 713-735.

[5]

Z. CAI, J.J. JONES, S.F. McCoRMicK, AND T.F. RUSSELL, Control-volume mixed finite element methods, Computational Geosciences, 1 (1997) 289-315.

[6]

Z. CAI, J. MANDEL, AND S.F. MCCORMICK, The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal. 28 (1991), 392-402.

[7]

S.H. CHOU AND P.S. VASSILEVSKI, A general mixed co-volume framework for constructing conservative schemes for elliptic problems, Math. Comp., 68 (1999).

[8]

S.H. CHOU, D.Y. KWAK, AND P.S. VASSILEVSKI, Mixed covolume methods for elliptic problems on triangular grids, SIAM J. Numer. Anal. 35 (1998), 1850-1861.

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[9]

T. IKED A, Maximum Principle in Finite Element Models for Convection-Diffusion Phenomena, Lecture Notes in Numer. Appl. Anal., Vol. 4, North-Holland, Amsterdam New York Oxford, 1983.

[10]

C. JOHNSON, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987.

[11]

R.D. LAZAROV, J.E. PASCIAK, AND P.S. VASSILEVSKI, Iterative solution of a combined mixed and standard Galerkin discretization methods for elliptic problems, (submitted to Numer. Linear Alg. Appl.).

[12]

M. Liu, J. WANG, AND N.N. YAN, New error estimates for approximate solutions of convection-diffusion problems by mixed and discontinuous Galerkin methods, Univ. of Wyoming, Preprint, 1997.

[13]

I.D. MISHEV, Finite volume methods on Voronoi meshes, Numerical Methods for Partial Differential Equations, 14 (1998), 193-212.

[14]

P.A. RAVIART AND J.M. THOMAS, A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods (I. Galligani and E. Magenes, eds), Lecture Notes in Math., Springer-Verlag, 606 (1977), 292-315.

[15]

P.A. RAVIART AND J.M. THOMAS, Primal hybrid finite element methods for second order elliptic equations, Math. Comp., 31 (1977), 391-396.

[16]

M. TABATA, A finite element approximation corresponding to the upwind finite differencing, Mem. Numer. Math., 4 (1977) 47-63.

[17]

C. WIENERS AND B. WOHLMUTH, Coupling of mixed and conforming finite element discretizations, American Mathematical Society, Cont. Math. 218 (1998), 547-554.

Numerical computation of 3D two phase flow by finite volumes methods using flux schemes

J.-M. Ghidaglia Centre de Mathematiques et de Leurs Applications ENS de Cachan and CNRS UMR 8635 61 avenue du President Wilson 94235 Cachan Cedex France jmgQcmla.ens-cachan.fr http://www.cmla.ens-cachan.fr/~jmg

ABSTRACT We propose here a general class of cell centered finite volume methods specially designed for the discretization of complex models of partial differential equations like those occuring in 3D two phase flow. After a brief introduction to these models (the so called averaged models in Eulenan formulation), we develop all the tools which are needed in order to arrive to a fully discrete scheme suitable for coding. Hence we discuss conservative systems, non conservative ones, time discretization, discretization of source terms, of diffusion operators, of boundary conditions, .. . We also briefly discuss non conformal meshes. We strongly rely on the concept of flux scheme which is, according to us, very well suited for the problems considered here. Key Words : Two phase flow, Flux schemes, Finite volumes, non hyperbolic convection, Source terms, Footbridges.

1. Introduction

Our goal in this contribution is to discuss a class of cell center finite volume methods, on unstructured conformal or non conformal meshes, designed for complex models (such a goal was also addressed by T. Gallouet [GAL 96]

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in the previous version of this symposium). We are mainly motivated by the computation of two phase flows occuring in several fields of the nuclear power industry (of course there are many other contexts where multiphase models occur), that is flow of water and its vapor. Models pertaining to this field encompass many difficulties, whose non exhaustive list is follows : (i) the solutions might develop strong gradients, (ii) the equations might involve stiff source terms, (iii) non conservative products might be present in the equations, (iv) the thermodynamics could be highly complex (e.g. phase transition), (v) the total convection operator might be non hyperbolic, (vi) in some part of the flow low Mach numbers can occur,

(vii) . . . (viii) and last but not least : there is no "universal model" to work with. Some of these features are present in one phase Computational Fluid Dynamics (Iph-CFD), other are typical in combustion but 2ph-CFD often involves toghether all of them. In this paper we shall propose a very general framework for the discretization of such systems. Generality is essential in this context according to (viii). And passing from one model to an other (which might implies even the variation of the number of scalar equations to be considered) should cause only minor changes in the code. Moreover one should ask almost no a priori conditions on the model (besides the fact that it is build upon physical considerations). One of the major by-product of this stategy is that a code derived according to this philosophy can be a tool for the validation of the physical models, and this point is one of the main issues in 2ph-CFD at the present time. Most of the results presented here were inspired by a very fruitful colaboration with the Departement Transfert Thermique et Aerodynamique (TTA/D R&D/EDF) and with the Service de Simulation des Systemes Complexes et de Logiciels (SYSCO/DRN/CEA).

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2. A short discussion of the models

We consider the motion of two non miscible fluids, like e.g. air and water or water and steam. Since we are concerned with macroscopic dynamics, we shall consider a continuum model. For single phase flows, except for very hight speed rarefied flows, Navier-Stokes Equations provide a hightly satisfactory continuous model. For two phase flows, a first approach could consists in writting for each phase a Navier Stokes model completed with jump relations across interfaces between the two fluids. This is of course possible for very very simple situations and cannot be envisaged in general. We shall consider here so called "two fluid" models (in Eulerian formulation) obtained throught an ad hoc averaging process. That is at each point in space and time, the modelization assumes that both fluids are present and a parameter (which is also to be determined) represents the proportion of each fluid. Such a model can be derived by an averaging process and we refer to e.g. Ishii [ISH 75], Ransom [RAN 89], Drew and Wallis [DRE 94] for more details. The models rely on the three usual balance equations (mass, momentum and energy) for each fluid. Denoting with subscripts v and / the quantities attached to one of the two fluids (e.g. density, velocity, energy,..) and by av, a/ the volume fraction of each fluid : av -f a/ = 1, these equations reads as follows (*£{»,/}). Balance of mass.

where p^ is the density of the fluid k, u^. the velocity of the fluid k and F^ the density of the mass transfert to the fluid k resulting from interfacial exchanges with F7; + T; = 0. Balance of momentum.

where P is the pressure in the mixture, Tk the viscous stress tensor, g the gravity field, Ik the volumic density of the momentum transfert to the fluid k resulting from interfacial exchanges (after substraction of the pressure contri-

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bution) and Balance of total energy.

where e^ is the specific internal energy of the fluid k, hk the specific enthalpy of the fluid k, &kQk the volumic heat transfered to the fluid k, QkQk the heat flux in the bulk of the fluid k, Hk the volumic density of the energy transfered to the fluid k resulting from interfacial exchanges with Hv + H/ = 0. Once this set of differential equations written, it remains to close the system by adding enought closure relations in order to obtain the same number of equations and independent unknowns. At the present time, as emphasized in the Introduction, it is an open problem. However by using various methods it is indeed possible to obtain, in definite physical contexts, some closed models from the equations above. These models will in general leave free a few number of constants that are to be determined by experiments (correlations). Hence the numerical modelisation of two fluid flows must be strongly coupled with experimental programs. Basic models We give below three typical models (that can be written in 1, 2 or 3 space dimensions). These models are very simple in the sense that they do not involve complicated physical correlations in their right hand side. Actually these models are used in the context of numerical benchmarks designed in order to check that the numerical method used is able to capture some of the important features of the flow considered and also in order to evaluate the numerical dissipation of the method. We refer to [MIM 99] for a collection of numerical benchmarks.

2.1. A 3 equation model This is the so called homogeneous 3 equation model that we write for the sake of simplicity in ID :

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Here p represent the density, u the average velocity, ur the relative velocity between phases, c the quality, L the latent heat and 3> a heat flux (source term). See also Example 1.

2.2. A 4 equation model This two-fluid model reads as :

Here p^ represents the density of the phase fc, u^ its velocity, p is the pressure, and a/5 denotes the volume fraction of the phase k, (av + at = I). The relative velocity is simply ur — uv — u^. Here, the source term is a drag force which is given by the correlation k = ^avpi^- in equations (9) and (10), where R^ is the bubble radius and CD the drag coefficient. 2.3. A 6 equation model In the previous two fluid model, no energy balance equations were written. This corresponds to isentropic flows. Here we want to consider the full set of balance equations (mass, momentum and energy) and this leads to a so called six equations model (which lead in 3D to a set of 10 scalar evolution partial differential equations). Let us consider the following system of equations :

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We still have to describe the thermodynamics of this "one-pressure" model. To this extent, we assume that two pressure laws are given, one for the "gas" and one for the "liquid" phase :

Here e^, denotes the internal phase energy and

3. The finite volume approach

The goal of this section is to describe the program that lead to the full discretization of the models of Section 2. We shall not present numerical results refering to the bibliography for that purpose : [ACT 99], [BOU 98], [GKL 96], [GKT 97], [GP1 99] , [HAL 98], [KUM 97], [PAS 99], [TAJ 96], [TAJ 98], [TKP 99]. In Section 3.1, we first address the conservative case which allows us to introduce the basics of the finite volume discretization. Then in Section 3.2, we introduce the characteristic flux approach. This lead us in Section 3.3 to the concept of flux schemes. In Section 3.4, we extend the previous analysis to non conservative systems while Section 3.5 deals with the time discretization. In Section 3.6 we put the emphasis on the discretization of forcing terms while Section 3.7 is devoted to the implementation of boundary conditions. In Section 3.8, we propose a convenient way of discretizing diffusion operators and finally in Section 3.9 we give some elements on the generalization to non conformal meshes.

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3.1. The conservative case Althought almost all the models encountered in two phase fluid flow are non conservative, we are going to deal first with conservative ones since they are convenient in order to introduce the basic notations. Let us consider a system of m balance equations :

where here Dosed m a nu-dimensional domain

maps

This equation is in practice).

Example 1 The homogeneous model (4) to (6) enters in this category. Assuming that a thermodynamic law p = P(p,e) is given, and when ur = 0, one recovers the usual Euler equations. In the case where ur ^- 0, we have also to prescribe the quality c = C(p, e) and in this case, we can write (4) to (6) into the form (18) provived we set v\ — p, v? = pu, vs = pE, F± = pu F} = p(u2 +c(l - c}ul] + p, F3! = p(uH + urc(l -c)(L + uur + (| - 2c)uj!)). Here E = e+||u| 2 , H = E-\-V-. It should be noticed that even when ur is a fixed constant and even for the physical laws of water, this system is not necessarily hyperbolic [TAJ 98]. We assume that the computational domain Q is decomposed into smaller volumes (the so called control volumes) K : Q, — UK^T^ and, in a first step, we assume that Q = U^gy-A' is "conformal" i.e. that it is a finite element triangulation of £1. In practice one can use triangles for nd = 2 and tetrahedrons for nd — 3. The cell-center finite volume approach for solving (18) consists in approximating the means

where vol(K) denotes the nd-dimensional volume of K and area(A) stands for the (nd — l)-dimensional volume of an hypersurface A Integrating (18) on K makes the normal fluxes, FQK, appear

where dK is the boundary of K, J'(cr) the unit external normal on dK and da denotes the (nd-l)-volume element on this hypersurface. Indeed, we have

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according to (18) :

The heart of the matter in finite volume methods consists in providing a formula for the normal fluxes FQK in terms of the {VL}L^T- Assuming that the control volumes K are polyhedra, as it is most often the case, the boundary dK is the union of hypersurfaces K D L where L belongs to the set A/"(/C), the set of L € T, L / /\, such that K fl L has positive (nd — l)-measure. We can therefore decompose the normal flux as a sum :

where (v,k.l points into L) :

Motivated by the case where (18) describes wave propagation phenomena, it is natural to look for an approximation of (23) in terms of vx(t) and v ^ ( t ) :

where $ is the numerical flux to be described. At this level of generality

where v G Mm and / : Mm H-> Mm: (i.e. (18) ) when nd = 1). We denote by A(v) the jacobian matrix ^' and observe that since A(v}^ = ^y' then according to (25),

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This shows that the flux f(v) is "convected" by A(v) (like v since we also have

Let us assume that (25) is smoothly hyperbolic that is to say : for every v there exists a smooth basis ( r i ( v ) , . . ., rm(v)) of R m made with eigenvectors of A(v] : 3\k(v) G M such that A(v)rk(v) = \k(v)rk(v). It is then possible to construct ( l i ( v ) , . .. , l m ( v ) } such that t A(v}lk(v] = \ k ( v ) l k ( v ) and lk(v)-rp(v) — &k,PThe numerical flux <J> represents the flux at an interface. Using a mean value p of v at this interface, we replace (26) by the linearization :

It follows that setting fk(v) = Ik(l^) ' f ( v ) , which is termed as the k-th characteristic flux, we have

Solving exactly this last equation (see also Remark 7), leads to the following formula :

Remark 1 Non hyperbolic convection operators In the complete physical models, the equations for the balance of momentum and energy (see (2) and (3)) involve terms with second order gradients like e.g. molecular and turbulent dissipation and diffusion. Here we are only discussing the connective part, i.e. terms with first order gradients, see Section 3.8 concerning the discretization of second order gradients. It might happen (this is very often the case when dealing with two phase flow, [STE 84]), that this convection operator is not hyperbolic (see also Example I). That is the matrix A(v) might have complex eigenvalues. When it is the case, we keep formula (29) unchanged but this time, \k(v) denotes the real part of the k-th eigenvalue of A ( v } . The physical meaning of this is that we still identify these numbers as physically meaningful wave speeds.

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3.3. Flux schemes We emphasize the fact that formula (29) has the following remarkable property : the numerical flux at an interface <£ is a linear combination of the fluxes f ( v ) and f ( w ) on both sides of the interface. This has led to the definition of flux schemes ([GHI 98]). Definition 1 The numerical flux <$ corresponds to a flux scheme when there exists a matrix U(v,w,K,L) such that

The extension of formula (29) to the multidimensional case is therefore given throught the following definition. Definition 2 The numerical flux of the "VFFC" method is obtained by formula (30) when we take :

where n(v,w]K,L) is a mean between VK and VL which only depends on the geometry of K and L :

Remark 2 In formula (31) the sign of a matrix M is the matrix which has the same generalized eigenspaces as M but whose eigenvalues are the sign ( —1, 0 or +1) of the corresponding eigenvalues of M. When the eigenstructure of M is not explicitely known, this matrix can be numerically computed in two different ways, . First one can use a numerical package that determines this eigenstructure. Second one can use a very efficient iterative method (a variant of the Newton-Schultz algorithm, see [ALO 99]) which produces a polynomial sequence of matrices converging rapidly to the matrix sgn(M). Remark 3 Other flux schemes Many other numerical schemes are flux schemes (see [GHI 98]). They have in general a very similar behavior as far as numerical accuracy is concerned. For example Roe's scheme is a flux scheme. For complex models like those considered here, determining Roe's average can be a

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difficult challenge while our VFFC scheme involve a simple geometric average. We note also that Stddke et al [SFW 97] have proposed also a flux scheme (for a particular system) with a simple average. Concerning Roe's scheme, Toumi [TOU 92] has derived very efficient procedure in order to use this scheme in the context we consider here. Later Halaoua [HAL 98] has generalized Roe's scheme to non hyperbolic convection operators. We particularly recommend the paper by Toumi, Kumbaro and Paillere [TKP 99] for a very nice review on these questions (see also [GLT 99]. Proposition 1 Combining (21), (22), (24), (30), (31) and (32), the finite volume approximation of (18) is the following system of o.d.e. 's :

3.4. The non conservative case Let us now consider a system of m balance equations which reads as :

and which therefore generalizes (18). The terms Cj(v)j£- are called non conservative products, and they introduce new difficulties both from theoretical and practical point of view. Often, such terms come from modelisation (e.g. approximations, closure relations,...) procedures and according to the way they are dicretized, one can obtain non physical solutions or even numerical divergence phenomena. Such terms are present in almost all the models encountered in two phase flow fluid dynamics and they represent one of the major difficulty in this area. Example 2 The two fluid model (7) to (10) enters in this category when k = 0 (otherwise we just have a source term on the right hand side of (34))- Here we take v\ = avpv, v? = otipi, vs = avpvuv, v4 = atptut, F/ = avpvuv, F\ — ottptut, Fg = av(pvul -\-p), F| — ai(piu^ + p) and the matrix 6*1(7;) is such that :

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Remark 4 Of course the form 34 is not unique since one can add an arbitrary function of v, G, to F and substract VvG(v] to the matrix C(v). Since the function F which appears has a physical meaning (the so called physical flux), its form is chosen on physical ground rather than on an abstract mathematical principle. For example the choice made in Example 2 is dictated by the property that when we have only one phase which is present (e.g. when av = 0) one recovers the classical fluxes of the Euler equation. Actually, other non conservative products can occur, and the general system of equations reads as :

Example 3 For the 3-dimensional case (nd = 3), we can recast system (11) to (16) under the form (35) with m = W ; v\ = avpv, v^ — (*tpt, t>3 = avpvUVii, V4 =

ttlPl^,!,

^5 = C*vpvUVt2, VQ = QlpiU^^, ^7 = CtvPvUVi3, V8 =

Q

tPiut,3, VQ — avpvEv and VIQ — aipeEt. Then one gets the F1- like in Example 2 (they are given expilcitely in [SOU 98]). Assuming that the matrix I + D(v) is invertible (which is the case in practice), we can transform easily (35) into (34). Hence we can only discuss the discretization of this last equation. In the one dimensional case, equation (34) reads as :

Denoting this time by J ( v ) the jacobian matrix

where A(v) = J ( v } ( I -+- C ( v ) J ( v ) formula (31) is replaced by

and

1

we have instead of (26),

). Hence, in the multidimensional case,

where

and

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Remark 5 Of course when (7 = 0 both formulas (31) and (38) agree. Now that we have a formula for computing the numerical flux at an interface K D L, we have to discuss the discretisation of the non conservative products. Integrating equation (18) on a volume control K, leads to the question of approximating the integral fK Y^]=i Cj(v)-j£- dx. Following [BOU 99] (see also [BOU 98]), we take :

where EK,L is the following m x m matrix :

Proposition 2 In the non conservative case, the finite volume approximation of (34) ^s the following system of o.d.e. 's :

3.5. Fully discrete schemes In the previous Section, we arrived to systems of o.d.e.'s ((33 and (41)). In order to actually compute the solutions, we have to perform a time discretization. This discretization can be either explicit or implicit. As it is well understood, the explicit discretization of the convective flux will lead to a stable scheme only under a Courant-Friedrichs-Lewy (C.F.L.) condition. This condition will impose on the time step a very restrictive constraint. Namely that this number must be smaller than Ax/c where Ax is a measure of the smallest length in the space discretization and c a measure of the largest speed of propagation appearing in the physical problem (note that in liquid and vapor flows this speed can be of the order of 2000 ms~1}. Hence when there are long transients, explicit methods are not effective in this context.

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We consider (34). The previous Section was devoted to the space discretization of this equation and we arrived to (41). The Euler explicit scheme for this equation reads :

As already said, this scheme is stable only for small time step. An alternative is then Euler implicit schemes. We solve the following equation for v^+l :

Althought of interest, this method can be too diffusive since all the characteristics are made implicit regardless the magnitude of the characteristic speed. We have proposed in Ghidaglia, Kumbaro, Le Coq and Tajchman [GKT 97] an alternative method in ID (see Remark 7) which has been generalized to the nd-D case by Boucker [BOU 98]. Definition 3 In the multidimensional case, the numerical flux of the "implicitexplicit VFFC" scheme is given by

where the m x m matrix

{1, 2, 3, 4} are defined as follows :

with n denned bv

the eigenelements (l,r and X) refer to the jacobian matrix the uj£ are given according to the following cases. (i) When 0

[

and

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When

(iv)

When

(v)

When

83

Above, &K,L denotes a characteristic lenght in K which depends also on L and K is a nondimensional number between 0 and 1 to be specified (see Remark 6). Remark 6 We can take for example

and K is determined thanks to the stability limit of the explicit scheme, see Boucker [BOU 98] (for example K w 0.3 for nd = 3y). Remark 7 the one dimensional case When nd = 1, formulas (44) to (51) are changed as follows :

where the m x m matrix

{1, 2, 3, 4} are defined as follows :

with n is defined by

and uj^ according to the following conditions.

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(i) When 0

(ii) When

(m) When

(iv) When

(v) When

In the case where the C.F.L. conditions :

are satisfied, formula (29) follows from these considerations.

3.6. On the discretization of source terms

Instead of (25), we want to solve

by the numerical scheme :

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Equation (63) can be obtained by integration of (62) on the rectangle [tn, tn+i] x [xj_i/2, Xj+i/2\ and this leads to the following form for E™ :

Actually, this formula is not suitable when the source term is large and one must modify (64) according to the expression of the numerical flux 9, where and

where U(-,-} is a matrix to be specified. Due to the fact that the numerical flux is not centered (upwind bias) one can observe large errors on the permanent solution under investigation. Let us introduce the notion of enhanced consistency as follows. Denoting by

we say that E^ satisfies, with respect to the numerical flux g, the enhanced consistency property when we have : if at some time-step, vj is such that

then v? +1 given by (63) must be equal to v^. This condition can be equivalently formulated as

if u" satisfies (67). In Alouges, Ghidagliaand Tajchman [AGT 99], the following result is shown. Theorem 1 Let (a,-) be given in the form (65) and denote by Un,L the matrix 3 ' 2 Uj+i(v^, y ?+i)- The enhanced consistency (i.e. (68)) will be satisfied if we discretize the forcing term S according to the following formula

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Observe that from the computational point of view, formula (69) induces almost no extra cost since Uj+± has already been computed in order to construct the numerical flux. In order to illustrate formula (69), let us show what it means in the case of a single linear equation vt + c vx = S where e.g. c > 0 and v G M. Here the usual upwind scheme amounts to take [/" = 1, i.e. a(v. w] = c v and (69) reads as :

3.7. On the boundary conditions So far we have not discussed the implementation of boundary conditions. This is a very important matter since they actually determine the solution. Let us consider the space discretization of system (18) by our cell centered finite volume method. We have found the sytem of o.d.e.'s (33). Of course this evolution equation is not valid when K meets the boundary of Q. When this occurs, we have to find the numerical flux $(VK, A", d£l). In practice, this flux is not given by the physical boundary conditions and moreover, in general, (18) is an ill-posed problem if we try to impose either v or F(v) • v on d£l. This can be simply understood on the following linearisation of this system :

where v represents the direction of the external normal on iacobian matrix :

is the

and v_ is the state around which the linearisation is performed. When (18) is hyperbolic, the matrix J^ is diagonalizable on IR and by a change of coordinates, this system becomes an uncoupled set of m advection equations :

Here the c^ are the eigenvalues of J_^ and according to the sign of these numbers, waves are going either into the domain Q (c/c < 0) or out of the domain H (c/j > 0). Hence we expect that it is only possible to impose p conditions on

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K fl dSl where p = ${k e {1,. . ., m) such that ck < 0}. Let us consider now a control volume K which meets the boundary dQ. We take v_ — vK and write the previous linearisation. We denote by x the coordinate along the outer normal so that (71) reads :

which happens to be the linearisation of the ID (i.e. when nd = 1) system. First we label the eigenvalues Cf~ (v_) of J^ by increasing order :

(i) The case p — 0. In this case all the informations come from inside £1 and therefore we take :

In the Computational Fluid Dynamics litterature this is known as the "supersonic outflow" case. (ii) The case p = m. In this case all the informations come from outside fi and therefore we take :

where $given are the given physical boundary conditions. In the Computational Fluid Dynamics litterature this is known as the "supersonic inflow" case. (iii) The case 1 < p < m — I. As already discussed, we need p informations coming from outside of f2. Hence we assume that we have on physical ground p relations on the boundary :

Remark 8 The notation gi(v) — 0 means that we have a relation between the components o f v . However, m general, the function gi is not given explicitely in terms of v. For example gi(v) could be the pressure which is not, in general, one of the components o f v .

88

Finite volumes for complex applications Since we have to determine the m components of <&(VK, K, d£l), we need m — p supplementary informations. Let us write them as

In general (78) are termed as "physical boundary conditions" while (79) are termed as "numerical boundary conditions". Then we take : where v is solution to (78)-(79) (see however Remark 10 and (86)). Remark 9 The system (78)-(79) for the m unknowns v^Gisamxm nonlinear system of equations. We are going to study its solvability, see Theorem 2. Let us first discuss the numerical boundary conditions (79). By analogy with what we did on an interface between two control volumes K and L, we take (recall that v_ = VK) '•

In other words, we set hf,(v) = lk(vK)-(F(v)-i/K)—lk(vK)-(F(vK)-VK)We have denoted by ( / i ( v ) , . . . ,/ m (v)) a set of left eigenvectors of J^ : tJ_^lk(p_) — Cklk(v_) and by ( r i ( u ) , . . . , rm(v_)) a set of right eigenvectors of J^ : J^rk(v_) = c/ c r/ t (v). Morover the following normalization is taken : lk(v_) • rp(v_) — $k,pAccording to Ghidaglia [GHI 99] we have the following result on the solvability of (78)-(79). Theorem 2 Assume that

With the choice (81) the nonlinear system (78)-(79) has one and only one solution for v — v_ sufficiently small. Remark 10 In practice, (78)-(79) are written in a parametrized manner. We have a set of m physical variables w (e.g. pressure, densities, velocities,...) and we look for w satisfying :

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and then we take : The system (83)-(84)-(85)

3.8.

is then solved by Newton's method.

On the discretization of viscous fluxes

Our aim is to introduce an algorithm for the dicretization of second order elliptic operators in the context of finite volume schemes. The technique consists in matching to a finite volume discretization based on a given mesh, a finite element volume representation on the same given mesh. An inverse operator is also built. This numerical algorithm, which was introduced in [GP1 99], (see also [GP2 99], [GP3 99] and [PAS 99], is based on footbridges between the finite volume representation of functions and a finite element one, both defined on the same mesh. We are going to give some details in the two dimensional case, refering to the aforementioned references for a more complete account and for the general ncf-dimensional case (which is in fact similar to the 2D case). Let £2 be the computational domain that we assume being triangularized : Q = \\ T. Let XT be the piecewise-constant characteristic function defined rer on each triangle T. For each edge A, let us define the piecewise-linear function (f>A such that

Similarly, to each u in £, we first construct v = HVEU in V solution of :

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The finite element representation allow to discretize easily a viscous operator thanks to a variational formulation and the footbridge HVE allow to resume to a finite volume discretization well suited for applying upwinding. But YlyE ° n^v 7^ Idy, then in the case of a zero diffusion parameter, the solution is modified by the use of the two footbridges. We construct another footbridge n from E into V such that II o MEV = Idy. For that purpose, it is sufficient that for each UEF in E to find VVF in V solution of the system :

Our first observations are that HEV is injective and that (89) has no solution in general. Let B be the N F x NE matrix such that Bja = fnXa

find

minimizing

is used to solve this system. Since the rank of B is NE, (90) has a unique t solution uv F — PVEUEF. Its value, ^BB) BbEF, has low computational t cost since BB is a sparse matrix and a Cholesky factorization can be applied. Hence we get PVE ° ^EV — Idy. 3.9. Non conformal meshes Non conformal mesh techniques were investigated in two situations : • when partitionning the whole domain into several sub-domains, allow a non conformal junction between meshes of adjacent sub-domains (see figure l.b); • use local refinments, keeping the structured type of the main discretization (see figure l.a). Flux schemes have interesting features in these situations :

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Figure 1: Non conformal mesh examples

First, they are based on a finite volume, cell-centered, discretization i.e. no need to interpolate node values). Second, the main part of the algorithm consists in computing numerical fluxes passing through interfaces. So, we have to build once at the beginning an explicit (unstructured cases) or an implicit (structured cases) list of interfaces. At each timestep, the main loop will use this list to compute numerical fluxes and put flux contributions to interface neighbours. For unstructured meshes, the same algorithm can be applied to conformal or non conformal meshes. For structured meshes with local refinments, a treelike structure has been tested with simulated recursive procedures (see figure 2 which represents the tree associated to figure l.a). We refer to Tajchman and Freydier [TAJ 96] for more information. 4. Bibliography [ALO 99]

ALOUGES F., Matrice signe et systemes hyperboliques, in Aspects theonques et pratiques de la simulation numerique de quelques problemes physiques, 249-257, Memoire d'Habilitation, Universite Paris-Sud, 1999.

[ACT 99]

ALOUGES F., GHIDAGLIA J.M. AND TAJCHMAN M., On the interaction of upwinding and forcing for nonlinear hyperbolic

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Figure 2: Tree representation of a local refinment

systems of conservation laws, Prepublication du CMLA and article to appear, 1999. [BOU 98]

[BOU 99]

BOUCKER M., Modehsation numenque multidimensionnelle d'ecoulements diphasiques liquide-gaz en regim.es transitoires et permanents : methodes et applications, These, ENSCachan, France, 1998.

BOUCKER M. AND GHIDAGLIA J.M., On the discretization of non conservative terms in the context of multidimensional flux schemes, in preparation.

[DRE 94]

DREW D.A. AND WALLIS G.B., Fundamentals of Two-Phase Flow Modeling, in Multiphase Science and Technology, Volume 8, Hewitt al Editors, Begell House, Inc., New-York, 1994.

[GAL 96]

GALLOUET T., Rough schemes for complex hyperbolic systems Proceedings of the first symposium on finite volumes for complex applications 15-18 July 1996, 1996, Editions Hermes Paris.

[GHI 98]

GHIDAGLIA J.M., Flux schemes for solving nonlinear systems of conservation laws, Proceedings of the meeting in honor of P.L. Roe, Chattot J.J. and Hafez M. Eds, Arcachon, July 1998, to appear.

Invited speakers [GHI 99]

[GKL 96]

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GHIDAGLIA J.M., On the numerical treatment of boundary conditions in the context of flux schemes, In preparation.

GHIDAGLIA J.M., KUMBARO A. ET LE COQ G., Une methode volumes-finis a flux caracteristiques pour la resolution numerique des systemes hyperboliques de lois de conservation, C.R.Acad. Sc. Pans, 1996, 322, I, 981-988.

[GKT 97]

. GHIDAGLIA J.M., KUMBARO A., LE COQ AND TAJCHMAN M., A finite volume implicit method based on characteristic flux for solving hyperbolic systems of conservation laws, Proceedings of the Conference on : Nonlinear evolution equations and in finite-dimensional dynamical systems (ShangaT, June 1995), Li Ta-Tsien Ed., 1997, World Scientific, Singapore.

[GLT 99]

[GP1 99]

Ghidaglia J.M., Le Coq G. and Toumi I., Two flux-schemes for computing two-phases flows throught multi-dimensional finite volume methods, NUclear REactor Thermal Hydraulics (NURETH 9), October 1999

GHIDAGLIA J.M. AND PASCAL F., Passerelles volumes finis - elements finis, C.R.Acad. Sc. Pans , I, 328, 711-716, 1999.

[GP2 99]

GHIDAGLIA J.M. AND PASCAL F., Passerelles volumes finis - elements finis, methodes et applications, Rapport EDF/DER/TTA/HT-33/99/002/A, Chatou, France, 1999.

[GP3 99]

GHIDAGLIA J.M. AND PASCAL F., Footbridges between finite volumes and finite elements, Article to appear.

[HAL 98]

HALAOUA K., Quelques solveurs pour les operateurs de convection et leurs applications a la mecamque des fluides diphasiques, These, ENS-Cachan, France, 1998.

[ISH 75]

ISHII M., Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris, 1975.

[KUM 97]

KUMBARO A., Methode VFFC, application au probleme du robmet de Ransom, Rapport EDF/DER/TTA/HT33/97/021/A, Chatou, France, 1999.

[MIM 99]

MlMOUNl S., Quelques cas tests numeriques pour la simulation des ecoulements diphasiques, Rapport EDF/DER/TTA/HT-33/98/031/B, Chatou, France, 1999.

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[PAS 99]

PASCAL F., Passerelles Volumes finis - elements finis, Proceed-

ings of the H eme seminaire sur les ecoulements compressibles et la mecanique des flmdes numerique, CEA Saclay, Janvier 1999. [RAN 89]

RANSOM V.H., Numerical Modeling of Two-Phase Flows, Cours de 1'Ecole d'ete d'Analyse Numerique - CEA-INRIAEDF, 12-23 juin 1989.

[SFW 97]

STADKE H., FRANCHELLO G. AND WORTH B., Numerical simulation of multi-dimensional two-phase flow based on flux vector splitting, Nucl. Eng. Design, 1997, 177, 199-213.

[STE 84]

STEWART H.B., WENDROFF B., Two-phase flows : models and methods, J. Comput. Phys., 1984, 56, 363-409.

[TAJ 96]

TAJCHMAN M. AND FREYDIER P., Methode Volumes Finis a Flux Caracteristiques. Application a un calcul bidimensionnel sur un maillage non conforme, Note, Departement Transferts Thermiques et Aerodynamique, Direction des Etudes et Recherches, Electricite de France, EDF HT-30/96/004/A, May 1996.

[TAJ 98]

TAJCHMAN M. AND FREYDIER P., Schema VFFC : application a I 'etude d'un cas test d'ebullition en tuyau droit representant le fonctionneraent en bomllotte d'un coeur REP. Note EDF HT-33/98/033/A, decembre 1998.

[TOU 92]

TOUMI I., A weak formulation of Roe's approximate solver, J. Comput. Phys., 1992, 102, 360-373.

[TKP 99]

TOUMI I., KUMBARO A. AND PAILLERE H., Approximate Riemann solvers and flux vector splitting schemes for twophase flow Note presented at the 30t/l VKI CFD Lecture Series, 8-12 March 1999.

The MoT-ICE: a new high-resolution wave-propagation algorithm based on Fey's Method of Transport

Sebastian Noelle Institut fur Angewandte Mathematik Universitdt Bonn Wegelerstr. 10 53115 Bonn, Germany [email protected] uni-bonn. de

ABSTRACT The numerical solution of multi-dimensional systems of conservation laws is dominated by up wind-schemes based on one-dimensional Riemannsolvers. Such schemes neglect the physical directions of wave-propagation and replace them by the grid-directions. One of the algorithmic alternatives is Fey's Method of Transport (MoT). Similarly to its one-dimensional fore-runner, the Steger- Warming-Scheme, the MoT suffers from an inconsistency at sonic points. Here we derive a new version of Fey's multi-dimensional flux-vector-splitting and a globally consistent second-order-accurate characteristic scheme based on Interface-Centered-Evolution, the MoT-ICE. Numerical experiments show the stability, accuracy and efficiency of the new scheme. Key Words: Systems of Conservation Laws, Multi-dimensional Flux-VectorSplitting, Fey's Method of Tranport.

1. Introduction Since the work of Godunov, Van Leer, Harten-Lax and Roe, the numerical solution of systems of hyperbolic conservation laws is dominated by Riemannsolver based schemes (see [HIR 90] for the classical references). These onedimensional schemes are extended to several space-dimensions either by using dimensional-splitting on cartesian grids or by the finite-volume approach on un-

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structured grids. For both approaches, convergence and error estimates have been established for multi-dimensional scalar conservation laws, see for example the results and references in [CCL 94, KNR 95, NOE 96, WN 99]. Naturally, there are no comparable results for multi-dimensional systems, since no existence and uniqueness of the p.d.e.'s is known in this case. The first systematic criticism of using one-dimensional Riemann-solvers for multi-dimensional gas-dynamics goes back to Roe himself [ROE 86]: the Riemann-solver is applied in the grid- rather than the flow-direction, which may lead to a misinterpretation of the local wave-structure of the solution. A description of a number of failings of exact and approximate Riemann-solvers for the two-dimensional Euler-equations of gasdynamics may be found in Quirk's paper [QUI 94]. LeVeque and Walder [LW 92] present difficulties of Godunov's scheme for strong two-dimensional Shockwaves arising in astrophysical flows and propose the use of rotated Riemann-solvers. In [ROE 91, NOE 94] Roe and Noelle study oscillations generated by dimensional-splitting-schemes for a prototype linear system. Since the mid-eighties, Roe, Deconinck, Van Leer and many others developed the so-called fluctuation-splitting schemes for the equations of gasdynamics. In these schemes the divergence of the multi-dimensional flux-vector (called fluctuation) is split according to a grid-independent wave model (see [VNL 92] for references). In [NOE 94], the author developed a fluctuationsplitting-scheme for a 2x2 model problem which cures the directional effects exhibited by dimensional-splitting-schemes. However, this fluctuation-splittingscheme seems to be very dissipative for non-stationary problems. Another multi-dimensional approach is the Corner-Transport-Upwind (CTU) scheme of Colella [COL 90]. Here waves are not only propagated to the neighbors which join a common side with a cell, but also to the corner cells. LeVeque's wave-propagation-algorithm, which forms the backbone of the publicly available software-package CLAWPACK, splits one-dimensional fluxes in the transverse directions and propagates them to the corner cells (see [LEV 97] and the references therein). In [BT 97], Billet and Toro introduce weightedaverage-flux (WAF) schemes with good multi-dimensional upwinding- and stability-properties. In this paper we follow the flux-vector-splitting approach, which was first proposed by Steger and Warming [SW 81] in one space-dimension. Instead of decomposing the divergence of the flux-vector, as in the fluctuation-splitting schemes of Roe et.al., the conservative variables and the flux-vector themselves are split (see [HIR 90], Ch.20.2). In his dissertation, Fey developed a multidimensional version of the Steger-Warming-scheme, and called it Method of Transport (MoT). This scheme integrates the acoustic waves over the entire

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Mach-cone (see also [LMW 97] for recent related progress). Subsequently Fey, Jeltsch et.al. simplified the MoT and expanded it in various directions (for references, see [FJMM 97, FEY 98a, FEY 98b]). The starting point of this work are the papers [FEY 98b, FJMM 97]. In these papers, the MoT takes the following form: Step 1. A multi-dimensional wave-model leads to a reformulation of the conservation law as a set of nonlinearly coupled advection equations. Step 2. At the beginning of each timestep, the system is linearized and decomposed into a set of scalar advection equations with variable coefficients. Step 3. The solution of each scalar advection equation at the end of the timestep is computed using a characteristic scheme. Step 4. The solution is projected back onto the conservative variables using the wave-model of Step 1. Our own contribution to Fey's method may be summarized as follows: Given a multi-dimensional wave-model, we simplify Step 2 by distinguishing clearly between linearization and decomposition error. This makes it possible to write down a general second-order-correction term for the decomposition error in a single line. Then we discuss the first-order version of Fey's characteristic scheme (Step 3) and show an inconsistency of the numerical scheme on the scalar level. Subsequently, we derive new first- and second-order characteristic schemes which are globally consistent. Due to the simpler second-ordercorrection term for the decomposition error and a particularly simple choice of numerical transport-velocities our second-order scheme needs only 2.2 times the CPU-time of the first order scheme, and is very efficient. Fey's transport algorithm might be called MoT-CCE, since his scalar scheme uses Cell-Centered-Evolution. We call our new scheme, which is based on Interface-Centered-Evolution, MoT-ICE. In Section 2, we discuss the multi-dimensional wave-models and the linearization and decomposition steps. In Section 3, we show the inconsistency of the MoT-CCE and present the MoT-ICE. In Section 4, we give numerical experiments. Finally, in Section 5, we discuss topics for future research. This is a report on work in progress. The proofs of the theoretical results in Sections 2 and 3 as well as several important implementation details in Section 4 will be given in the forthcoming paper [NOE 99]. The author would like to thank Dr. Michael Fey and Christian von Torne for stimulating discussions on the Method of Transport. This work was supported by DFG-SPP ANumE. The hospitality of Dr. Fey, Prof. Jeltsch and his group during a visit at ETH Zurich is gratefully acknowledged. 2. Decomposition of multi-dimensional hyperbolic systems into scalar advection equations In this section, we recall Fey's advection form and give a general secondorder-accurate linearisation and decomposition of systems which can be written in advection form.

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2.1. A general framework for multi-dimensional flux-vector-splitting Consider a multi-dimensional system of n conservation laws in d space dimensions,

where U : Rd x R+ ->• Rn is the state-vector and f : Rn -> R n x d the fluxvector. The following framework for multi-dimensional flux-vector-splitting (FVS) has been developed by Fey, Jeltsch and collaborators [FJMM 97]: Definition 1 A wave-model for (1) is a set of L > 1 mappings

which satisfy the consistency conditions

and

Given a wave-model, the conservation law (1) may be rewritten in the following advection form:

Wave-models for the wave-equation, the equations of isentropic and non-isentropic gas dynamics and the equations of ideal magneto-hydrodynamics may be found in [FJMM 97]. Common to all of them is a finite set of acoustic waves which approximate the Mach-cone (typically four waves in two spacedimensions). For the Euler-equations one adds an entropy-wave, and for the MHD-equations, the two Alfven and possibly some slow magneto-acoustic waves. 2.2. Decomposition into scalar advection equations Let us see how the advection form can be used when advancing the solution for a single timestep k — At. Ideally, one would like to set each summand in (4) to zero seperately. As we shall see below, this will lead to an error of O(k2) during each time-step, limiting the overall accuracy of the scheme to first-order. The following linearisation will be helpful in order to obtain a second-orderaccurate decomposition:

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Lemma 2 (linearisation) Let U be a smooth solution of (1), k = At > 0. Let

satisfy

and

Then

The proofs of this and the following results will appear in [NOE 99]. Note that the approximate transport velocities a/ are now prescribed coefficients which depend on space but not on time. System (5) is still nonlinear in V. It is, however, linear in the components Sj(V), since V = 53j=1 Si(V). Next let us decompose the linearized advection form (5) by setting each summand to zero seperately: Let

satisfy

Then

where

is given by

Therefore, this naive ansatz leads to a first-order decomposition error. Taking a closer look at the leading part of the error, we note that

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since

This can be used in order to derive a second-order-accurate decomposition of (5) respectively (4). Theorem 3 (second-order-accurate decomposition) Let

satisfy

Then For the Euler equations and the shallow-water equations, Morel, Fey and Maurer have also derived a second-order-accurate decomposition into linear advection equations. Instead of evaluating the velocities at the half-timestep, as done in Lemma 2, they freeze them at the original timestep. As a consequence, the linearization- and decomposition-errors are not seperated, and the secondorder-correction terms become more involved. 3. Solving the scalar advection equations by characteristic schemes The linearisation and decomposition given in the previous section leads us, at the beginning of each timestep, to a set of scalar transport equations of the form

For the rest of this section, we consider the velocity-vector a : Rd x R+ —>• Rd to be a given function of x and t. The function (p : Rd x R+ —>• R is the unknown, and initial data are prescribed at t = tn := nk,

These equations may be solved by introducing characteristics z(r;x,t) by

z : R+ x Rd x R+ ->• Rd z(t;x, £) = x 0 T z(r;x,t) = a(z,r). Since the flux-vector

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Figure 1: Backwards characteristic transport of cell Kij. The region bounded by the curved lines is z(tn;Kij,tn+i), and the dashed lines are backwards characteristic curves issuing from the corners of cell K^ at time tn+\.

Lemma 4 For all K C Rd, t, r € R+

3.1 Characteristic schemes We would like to use Lemma 4 to construct numerical methods for solving (6). For simplicity, we restrict the analysis from now on to two spacedimensions, and use the notation x — (x, y) G R2 for the space variable and a = (a, b} e R2 for the velocity field. Let KIJ C R2 be the cells of a uniform cartesian grid with mesh-size h — Ax = Ay, and recall that A; = At is the timestep. Due to Lemma 4,

Note that Ki>j> fl z(tn;Kij,tn+i) is that part of KVJI that will be mapped to Kij by the characteristic flow from time tn to time tn+i (compare Figure 1). Definition 5 A characteristic scheme for (6) is given by

where (f>n is a piecewise smooth spatial reconstruction of <£>(•, tn), and K*L, an approximation of KVji n z(tn; Kij, tn+i).

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For the piecewise smooth spatial reconstruction of (p(-,tn) we will either use piecewise constant or piecewise linear functions. We will label the corresponding schemes P0 resp. PI. For the moment, let us consider schemes using piecewise constant reconstructions and focus on the approximation of the characteristic flow. 3.2 Cell-Centered-Evolution: the MoT-CCE In a series of papers (see [FEY 98b, FJMM 97] and the references therein), Fey and collaborators have used the following approximation of the characteristic flow, which we will call Cell-Centered-Evolution: In each cell Ki>ji, consider the local characteristic flow defined by

Then set In one space-dimension this leads to the following algorithm:

where A := k/h and the subscripts ± denote the positive resp. negative part of a quantity. Fey calls his scheme MoT ("Method of Transport"), and therefore we will use the acronym MoT-CCE-PO. Let us discuss the consistency of the MoT-CCE-PO with the differential equation (6). If a is constant, then (7) coincides with the first-order upwind scheme. For variable coefficients the situation is more complex: Example 6 Let us consider the case

corresponding to a compressive wave

For constant initial data,

the exact solution remains constant in space but grows exponentially in time,

The approximate solution produced by the MoT-CCE-PO is

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As k tends to zero with X fixed, this solution converges to

Thus (7) is inconsistent with the differential equation (6) at the "sonic" point x — 0, where the transport velocity a changes sign. The cell-centered-evolution leads to an inconsistent approximation of V • a, which takes the value 2 instead of 1. An analogous inconsistency occurs for the case of an expansive wave, V • a > 0. We omit the details. This difficulty was already described by Steger and Warming [SW 81] in 1981: the numerical flux produced by their splitting is not continuously differentiable at sonic points for the equations of gas-dynamics. This results in so-called "glitches" at sonic points. Subsequently, Van Leer developed a splitting with continuously differentiable fluxes (compare [HIR 90], Ch. 20.2.3). In her dissertation, Morel [MOR 97] also observed glitches at sonic points for two-dimensional shallow-water-computations carried out with the MoT-CCEPO, and generalized the Van-Leer-flux-vector-splitting to two space-dimensions in order to remove these numerical artefacts. However, this slows down the algorithm considerably, and Morel herself remarks that her method does not seem to be generalizable to second-order-accuracy. Finally, let us remark that in Example 6 above, the MoT-CCE-PO diverges in the L°°-norm, but converges in I/1. Moreover, this inconsistency does not occur when one approximates the velocity field a by piece wise linear functions. 3.3 Interface-Centered-Evolution:

the MoT-ICE

In the following we will derive a consistent alternative to the cell-centeredevolution discussed above. Since the new scheme will approximate the transport velocities at the interfaces between the cells, we will call it MoT-ICE for Interface-Centered-Evolution. Instead of using a cell-centered approximation of the local characteristic flow to decompose Kij, we now define auxiliary transport velocities an 1 . on + 2 '3 the vertical interfaces l

resp. bn.. ! on the horizontal interfaces J

J-T

2

between the cells. Let A = k/h be fixed and suppose that the CFL-condition

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Figure 2: Auxiliary transport velocities and approximate characteristic decomposition of a two-dimensional cell KIJ via Interface-Centered-Evolution.

is satisfied. In order to approximate the backwards characteristic flow for one timestep, we shift the vertical interfaces ^ j+i horizontally by (—k a" i .) and the horizontal interfaces JIJ+L vertically by (—k bn.

L ).

Depending on the

signs of the velocities a and 6, each cell KIJ will be divided into one up to nine sub-rectangles, leading to a natural subdivision

(compare Figure 2). For example,

and so on.

3.3.1. The MoT-ICE-PO In order to define the MoT-ICE-PO, we approximate (f> by piece wise con stants and define the auxiliary transport velocities by

where the remainders ai+i^ and fc^j+i are supposed to be lipschitz-continuous. Theorem 7 Suppose that \ — k/h is fixed, and that the CFL-condition (8) is satisfied. Then for any given smooth velocity field a : R2 x R+ —> R2, the MoT-ICE-PO is consistent of order one with the differential equation (6).

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Note that the theorem states global first order consistency. The difficulty is to control the truncation error near points where the characteristic velocities change sign. See [NOE 99] for the full proof. 3.3.2. The MoT-ICE-Pl Let us next construct a scheme based on piecewise linear reconstructions, the MoT-ICE-Pl. We have to define the auxiliary transport velocities at the interfaces and the piecewise linear reconstruction of the solution (p. We denote the piecewise linear reconstruction of R\ For ( x , y ) G Kij let

A rather involved computation leads to the following choice of auxiliary transport velocities at the interfaces which guarantees second order consistency for smooth solutions:

where a and b are supposed to be lipschitz-continuous. Let us give a geometrical interpretation of these terms: the factors (a 4- fat) resp. (b + |&t) simply reflect the fact that we wish to evaluate the transport velocities a resp. b in the center of the space-time interfaces II+LJ x [£n,*n+i] resp. Jij+L x [£ n ,t n+1 ] in order to achieve second-order accuracy in time. The factor (1 — |(az -I- 6 y )) which appears both in a and 6, is an approximation of

at t = k/1. This approximates the evolution of (p along the characteristics z(r;x,i), where (6) gives

The correction terms f (&&) y resp. f (ab)x account for fluxes across corners of the grid, see [NOE 99] for a more detailed discussion. Let us now state the consistency of the MoT-ICE-Pl: Theorem 8 Let a : R2 x R+ —> R2 be a given smooth velocity field and let (p : R2 x R+ —>• R be a smooth solution of (6). Suppose that there are lipschitz-continuous functions ^p"x and ^ such that the piecewise linear reconstruction (pn defined in (9) satisfies

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Let the auxiliary transport velocities ai+i j and bij+i be given by (10) (11), let A = k/h be fixed and suppose that the CFL-condition (8) holds. Then the MoT-ICE-Pl is consistent of order two with the differential equation (6). Once more, we refer to [NOE 99] for the proof, which requires a careful study of the truncation errors of the piecewise linear reconstruction (pp_. Let us stress again that we do obtain second-order consistency at all points, including those where the transport-velocities change sign.

4. Numerical experiments

In this section we present numerical experiments which confirm the accuracy and stability of the new MoT-ICE for smooth and discontinuous solutions. For the piece wise-linear reconstruction, we choose a central version of the WENO (Weighted Essentially Non-Oscillatory) reconstruction [JS 96]. Details may be found in [NOE 99]. 4.1 Scalar advection with periodic

coefficients

In order to illustrate the failure of consistency of the MoT-CCE-PO (cellcentered evolution) and the consistency of the new MoT-ICE (interface-centered evolution), we consider the one-dimensional scalar advection equation

over the interval [—1,1] with

and

Note that at x = 0, v?(0,£) satisfies the ordinary differential equation

so

Similarly, and these are the maximum and minimum of the exact solution. We compute the solution at time T — loe(2)/7r, so

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Table 1: EOCs for one-dimensional advection with periodic coefficients

ix 40 80 160 320 640 ix 40 80 160 320 640 ix 40 80 160 320 640

MoT-CCE-PO EOC LOQ Lj. 3.439066e-01 5.456309e-02 3.210264e-02 0.77 4.215299e-01 1.731594e-02 0.89 4.617376e-01 8.996724e-03 0.94 4.813319e-01 4.587514e-03 0.97 4.908176e-01 MoT-ICE-PO EOC Li LOO 8.834546e-02 3.075886e-02 1.599486e-02 0.94 4.741382e-02 8.114853e-03 0.98 2.471667e-02 4.094363e-03 0.99 1.263470e-02 2.055908e-03 0.99 6.379015e-03 MoT-ICE-Pl with WENO-limiter EOC Li ^00 5.123679e-03 2.193228e-03 2.04 1.532804e-03 5.330401e-04 1.281400e-04 2.06 3.878244e-04 2.993688e-05 2.10 7.604266e-05 5.934229e-06 2.33 1.523297e-05

EOC -0.29 -0.13 -0.06 -0.03 EOC 0.90 0.94 0.97 0.99 EOC 1.74 1.98 2.35 2.32

We choose a CFL-number of 41og(2)/yr, roughly 0.88. In Table 1, we list the experimental orders of convergence (EOCs) both with respect to the Ll and L°° norm. In the L^ncrm, the EOC of the MoT-CCE-PO starts at 0.77 and increases towards 1. However, the method diverges in L°°, as it should have been expected from Example 6 in Section 3.2. Contrary to that, the MoT-ICE converges uniformly (i.e. in L1 and I/00) to the expected orders. Comparing the L1 errors of the MoT-CCE-PO and the MoT-ICE-PO (piecewise constant reconstructions) , one sees that the MoT-ICE converges with a better rate (especially on the coarser grids) and produces roughly half the error of the MoT-CCE. The convergence rates of the MoT-ICE-Pl (piecewise linear reconstructions) are even better than 2 on the finer grids, both for the unlimited and the limited version (we omit the table for the unlimited scheme). The error of the scheme using WENO-limiter is only slightly larger than that of the unlimited scheme and it is of course orders of magnitude smaller than that of the first order MoT-ICE-PO.

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4.2 Rotating Smooth Hump Our next test problem is the two-dimensional scalar equation (6) with

in the domain [—1,1]2. Note that on this problem, the MoT-CCE-PO and the MoT-ICE-PO will produce identical results, since ax = by = 0, so

and

We compute one full rotation, i.e. T = TT, and fix A = At/Ax = Tr/6. Assuming that the maximal transport velocity is \/2, this corresponds to a CFL-number of roughly 0.74. First, we consider smooth initial data. Let XQ := 0.5, yo := 0, TO := 0.3,

Table 2: EOCs for smooth rotating hump. ix 40 80 160 320 640 ix 40 80 160 320 640 ix 40 80 160 320 640

MoT-ICE-PO EOC EOC Height Li LOO 9.528519e-01 0.287 7.129756e-01 0.454 6.479957e-01 0.56 5.4592186-01 0.39 3.994583e-01 0.70 3.690224e-01 0.631 0.56 2.277596e-01 0.81 2.228099e-01 0.73 0.777 1.228174e-01 0.89 1.239005e-01 0.85 0.876 MoT-ICE-Pl with unlimited central differences EOC EOC Height Li Loo 0.794 2.596250e-01 2.0565206-01' 2.24 4.491245e-02 2.20 0.961 5.505040e-02 1.091899e-02 2.33 8.892373e-03 2.34 0.995 2.426323e-03 2.17 1.967944e-03 2.18 0.999 5.822946e-04 2.06 4.673955e-04 2.07 1.000 MoT-ICE-Pl with WENO-limiter EOC EOC Height Li Loo 4.351708e-01 0.525 4.749751e-01 0.767 1.201747e-01 1.86 2.327933e-01 1.03 3.337591e-02 1.85 1.018161e-01 1.19 0.898 0.960 6.350393e-03 2.39 4.022132e-02 1.34 0.988 1.030164e-03 2.62 1.230235e-02 1.71

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and otherwise.

For the MoT-ICE-PO the EOCs increase slowly towards unity both in L1 and L°° (see Table 2). The error is very large on the coarser grids, and the convergence is initially slow. This is natural, since the initial data are not well resolved on the coarse grids, where they appear rather as a sharp peak than a smooth hump. For the MoT-ICE-Pl with unlimited central differences the EOCs are better than two both in L1 and L°° and they converge towards two as the grids are refined. For the MoT-ICE-Pl with WENO-Limiter, the EOCs in L1 are slightly below two in the beginning. However, as the grid is refined, the EOCs increase drastically and well beyond two. In L°°, the EOCs start slightly above unity on the underresolved coarse grids, but show a similar dramatic increase as the grids are refined. In our experience, this behavior is typical for the central WENO-limiter.

4.3 Rotating Cylinder Next, we consider the rotating cylinder:

othrewise

Since the solution is discontinuous, we only give the experimental orders of convergence in L1 (see Table 3). We also display the maximal height of the cylinder, in order to see if it is excessively smeared, or whether there are overshoots in the numerical solution. For the MoT-ICE-PO, the EOCs tend towards 0.5 as expected for a linear problem and a scheme based on piecewise constant reconstructions. The maximal height of the cylinder increases towards 1.0 as the grid is refined. For the unlimited MoT-ICE-Pl (we omit the table), the EOCs are decreasing from 0.78 towards 0.70. As should be expected for a

Table 3: EOCs for rotating cylinder. ix 40 80 160 320 640

MoT-ICE-PO EOC Height Li 7.012658e-01 0.868 5.046964e-01 0.47 0.968 3.618134e-01 0.48 0.990 2.583896e-01 0.49 0.995 0.997 1.837913e-01 0.49

MoT-ICE-Pl, WENO-limiter EOC Height Li 3.758964e-01 0.993 1.000 2.278516e-01 0.72 0.70 1.000 1.398394e-01 8.588859e-02 1.000 0.70 5.270885e-02 0.70 1.000

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Figure3: Rotating cylinder. MoT-ICE-PO and MoT-ICE-Pl, 160 x 160 points.

discontinuous solution computed with unlimited piecewise-linear reconstructions, there is an overshoot of 9 to 11 percent of the height of the cylinder. The MoT-ICE-Pl with WENO-limiter converges at rate 0.70, produces no overshoots and only slightly larger L1-errors than the unlimited scheme. As can be seen from the maximal height especially on the coarser grids, the computation is much less smeared than the one with the MoT-ICE-PO. The error on the finest grid is a factor 3.5 smaller for the limited MoT-ICE-Pl than for the MoT-ICE-PO. Figure 3 shows that the cylindrical shape of the solution is nicely preserved by both versions of the scheme. 4.4 Shallow-Water-Equations In her dissertation [MOR 97], Morel reports glitches at sonic points for a radially symmetric explosion for the shallow-water equations computed with the MoT-CCE-PO. Results using the new MoT-ICE, both PO and PI, show no such glitches, and produce almost perfectly radially symmetric solutions (see Figure 4). From these pictures, the results of the MoT-ICE-Pl seem to

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Figure 4: Explosion problem for the shallow water-equations. MoT-CCE-PO, MoT-ICE-PO, MoT-ICE-Pl. Grid of 160 x 160 Points. Left Column: 25 Contours of water-height. Right Column: cross-sections along the z-axis "+" and the diagonal "x".

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be of the same quality as those computed by Morel using CLAWPACK (see [NOE 99] for crucial implementation-details for the MoT-ICE-Pl for systems of conservation laws).

4.5 Comparison of

efficiencies

Let us give a first comparison of efficiencies. Morel reports that the MoTCCE-PO (which is inconsistent at sonic points), Van Leer's flux-vector-splitting and CLAWPACK T1'0 (using the first-order Roe-solver without transverse wave-propagation) all use the same amount of cpu-time (say one time unit). CLAWPACK T1'1 takes 1.5 units, and the first-order fix at sonic points proposed by Morel 2.9 units. Our preliminary experience with the MoT-ICE is the following: the MoT-ICE-PO, which is consistent at sonic points, takes 0.9 units and is hence as fast as standard first order schemes. The MoT-ICE-Pl takes 2.2 units, which is the sames as the second order CLAWPACK T 2 ' 2 . This compares favorably with the MoT-CCE-Pl, which is consistent at sonic points, but needs 10.5 units of cpu-time. 5. Discussion The new Method of Transport with Interface-Centered-Evolution (MoTICE) is consistent at sonic points, second-order-accurate for smooth solutions and nonoscillatory at discontinuities. In all test-calculations carried out by the author so far it produces high-resolution approximations of multi-dimensional phenomena with almost no grid-orientation effects. It needs about the same CPU-time as a second order version of LeVeque's wave-propagation algorithm CLAWPACK which, according to numbers given by Morel, is 4 to 5 times faster than the original second-order-accurate MoT-CCE-Pl. In ongoing joint work with von Torne, we plan to develop the MoT-ICE into a fully adaptive code, capable of handling general geometries. The multidimensional core of the Method of Transport are Fey's wavemodels. Both for theoretical and practical reasons, it would be desirable to establish a systematic approach for their derivation. Are there wave-models for any hyperbolic system of conservation laws, or do they require special symmetries, which are satisfied by the fluid-dynamical equations for which wavemodels have been found so far? Which properties should a good wave-model have, besides the formal consistency-requirements (2)-(3)? Can one analyze the interplay between properties of the wave-model and the stability and efficiency of the resulting wave-propagation algorithm? Let me make one final remark: I would very much like to see a more widespread continuation of the "Great Riemann-Solver Debate" started by Quirk [QUI 94]. Are the failures of Riemann-solver-based schemes reported by Quirk of a cosmetic nature, visible only in the "picture-norm", or can they

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be quantified? Do the schemes diverge or converge when the grid is refined? How do the so-called "genuinely multi-dimensional" schemes perform in these situations? 6. Bibliography For lack of space, we do not attempt to give a well-balanced (much less a comprehensive) bibliography. However, the following list should contain lots of useful information: [BT 97]

BILLET, S. AND TORO, E., On WAF-type schemes for multidimensional hyperbolic conservation laws. J. Comput. Phys. 130 (1997), 1 -24.

[CCL 94] COCKBURN, B., COQUEL, F. AND LfiFLOCH, P., Convergence of the finite volume method for multidimensional conservation laws. SIAM J. Numer. Anal. 32 (1995), 687 - 705. [COL 90] COLELLA, P., Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 87 (1990), 171 - 200. [FEY 98a] FEY, M. Multidimensional upwinding. I. The method of transport for solving the Euler equations. J. Comput. Phys. 143 (1998), 159 - 180. [FEY 98b] FEY, M. Multidimensional upwinding. II. Decomposition of the Euler equations into advection equations. J. Comput. Phys. 143 (1998), 181 - 199. [FJMM 97] FEY, M., JELTSCH, R., MAURER, J. AND MOREL, A.-T., The method of transport for nonlinear systems of hyperbolic conservation laws in several space dimensions. Research Report No.97-12, Seminar for Applied Mathematics, ETH Zurich (1997). [HIR 90]

HIRSCH, C., Numerical computation of internal and external flows. Vol.2, Wiley-Interscience 1990.

[JS 96]

JIANG, G.-S. AND SHU, C.-W., Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126 (1996), 202 - 228.

[KNR 95] KRONER, D., NOELLE, S. AND ROKYTA, M., Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. Numer. Math. 71 (1995), 527-560. [LEV 97] LEVEQUE, R.J., Wave Propagation Algorithms for Multidimensional Hyperbolic Systems. J. Comput. Phys. 131 (1997), 327 - 353.

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[LW 92]

LEVEQUE, R.J. AND WALDER, R., Grid alignment effects and rotated methods for computing complex flows in astrophysics, eds. J.B. Vos, A. Rizzi and I.L. Rhyming, Proceedings of the Ninth GAMM Conference on numerical methods in fluid mechanics. Notes Num. Fluid. Mech. 35 (1992), 376-385.

[LMW 97] LUKACOVA-MEDVIDOVA, M., MORTON, K. AND WARNECKE, G. Evolution Galerkin methods for hyperbolic systems in two space dimensions. Report 97-44, Univ. Magdeburg, Germany (1997). [MOR 97] MOREL, A.-T., A genuinely multidimensional high-resolution scheme for the shallow-water equations. Dissertation, ETH Zurich Diss. No. 11959 (1997). [NOE 94] NOELLE, S., Hyperbolic systems of conservation laws, the Weyl equation, and multi-dimensional upwinding. J. Comput. Phys. 115 (1994), 22- 26. [NOE 96] NOELLE, S., A note on entropy inequalities and error estimates for higher order accurate finite volume schemes on irregular families of grids. Math. Comp. 65 (1996), 1155 -1163. [NOE 99] NOELLE, S., The MoT-ICE: a new high-resolution wavepropagation algorithm for multi-dimensional hyperbolic systems of conservation laws based on Fey's Method of Transport. To be submitted for publication, 1999. [QUI 94]

QUIRK, J., A Contribution to the Great Riemann Solver Debate. Int. J. Numer. Meth. Fluid Dyn. 18 (1994), 555 - 574.

[ROE 86] ROE, P., Discrete models for the numerical analysis of timedependent multidimensional gas dynamics. J. Comput. Phys. 63 (1986), 458 - 476. [ROE 91] ROE, P., Discontinuous solutions to hyperbolic systems under operator splitting. Numer. Meth. Part. Diff. Eq. 7 (1991), 207. [SW 81]

STEGER, J. AND WARMING, R., Flux vector splitting of the inviscid gas-dynamic equations with applications to finite difference methods. J. Comput. Phys. 40 (1981), 263 - 293.

[VNL 92] VAN LEER, B., Progress in multi-dimensional upwinding. ICASE Report 92-43 (1992). [WN 99]

WESTDICKENBERG, M. AND NOELLE, S., A new convergence proof for finite volume schemes using the kinetic formulation of conservation laws. Accepted for publication in SIAM J. Numer. Anal. (Feb. 1999).

Numerical Analysis

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Error estimate for a finite volume scheme on a MAC mesh for the Stokes problem Philippe Blanc CMI, Universite de Provence Technopole de Chateau Gombert, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France email: [email protected]

Abstract We consider a finite volume scheme on MAC mesh for the Stokes equations. Under regularity assumptions on the solution, we prove an error estimate of order 2 in the L2 norm for the pressure and Hl norm for the velocity. Keys words

Finite volume scheme, MAC mesh, Stokes equations.

1. Introduction In this paper, we study some error estimate for the velocity in norm H1 and for the pressure in the L2 norm for the Stokes equations, discretized by a finite volume method on the "Marker and Cell" (MAC) mesh. This mesh was introduced by Harlow and Welch [HAR 65] in the middle of the sixties. Since then, the convergence of different schemes constructed on the MAC mesch has been studied for several methods. One of the first results for the convergence of the MAC scheme was found by Porshing [FOR 78] in 78. More recently in 1997, Shin and Strikwerda found an Inf-Sup condition [SHI 97] and an error estimates [SHI 96] for finite difference approximations. Two other methods are used on a MAC mesh: the mixed finite element method and the finite volume method. An example of mixed finite element method is given by Farhloul and Fortin in an article [FAR 97] in 1997. And in 1998, Han and Wu [HAN 98] proved a first order error estimate for both the velocity and the pressure. There exist several finite volume methods, some examples have been given, in 1983, in the book of Peyret and Taylor [PEY 83]. One of these has been studied by Nicolaides [NIC 92] in 1992 : the covolume method. This method

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allows the use of some results proved for "div-curl" systems. And in 1997, Nicolaides and Wu [NIC 97] proved a first order error estimate for the velocity and the pressure. In 1998, Chou and Kwak [CHO 98] introduced a covolume method for the generalized Stokes problem, and proved a first order estimate for the velocity and the pressure, provided that the exact velocity is in H2 and the exact pressure is in Hl. We choose here to directly discretize the velocity and pressure in the Stokes equations by a finite volume method, which we describe in the second part. Finally in the third part, we will prove a second order error estimate for the velocity and the pressure, provided that the exact velocity is in C3 and the exact pressure is in C2. 2. The finite volume scheme The purpose of this work is the study of a finite volume scheme for the discretization of the following Stokes problem:

where v is the viscosity, (w, v) is the velocity, p is the pressure and (/, g) is the given force. And we assume: Assumption 1 (i) (H) (Hi)

17 is an open bounded polygonal convex subset of M2, v > 0, /<EL 2 (17) and g € L2 (17)

We consider a MAC mesh on 17 (cf figure 1) satisfying: Assumption 2 There exists £ > 0 such that for all i and all j, we have:

(i) (ii) (Hi) (iv)

hi > C/i, kj > C/i, /ii+i/2 > (h, kj+i/2 > C^-

Numerical analysis

Figure 1: MAC mesh Then we write the finite volume scheme:

where

and for all

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and for all

This scheme [2]-[6] has a solution, which is unique up to an additive constant for the pressure (see e.g. [EGH 97]).

3. Error estimates

3.1. Notations Let (u,v,p) be the solution of the Stokes equation satisfying:

and we assume: Assumption 3 ue C 2 (O), veC2(fy

andpeCl(ti).

Then we denote: Notation 1

Let (uT,vT,pT) be the solution of the scheme [2]-[6] satisfying:

then we denote the error: Notation 2

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Remark 1 We have:

Notation 3

For a mesh T, XC'7") denotes the space of functions which are constant on each cell o f T .

3.2. Error estimates We prove here the following error estimate: Theorem 3.1

Under the assumptions 1 and 2, ifu G C 2 (fJ), v G C2(Cl), andp G Cl(Cl}, there exists C depending only of the exact solution (u, v,p) and fi such that:

Futhermore ifu € C 3 (fi), v € C*3((7), p G C 2 (fJ) and if tie mesh is uniform, then we have:

where ||e"||i)Tu and ||e^||i )Tv are the errors for the velocity in the discrete HQ norm (see [EGH 97] or [EGH 99] for the exact definition) and ||e£|| is the error for the pressure in the L2-norm. 3.2.1. Error estimate for the velocity Lemma 3.1

Under the assumption 1 and 3, and with the notations 2, there exists C\ depending only on (u, v,p] and on fi such that, for all a > 0, we have:

Furthermore ifu G C 3 (fJ), v G C3(Cl), p G C2(Cl) and if the mesh is uniform, then we have:

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To prove the first part of this lemma, we replace (ui+-\./2,j^vi,j+i/2jPi,j) m the scheme by (e"+1/2 - ' e i 7+1/2' e ??)> then we introduce some second order terms. Then we multiply each equation respectively by eV, 1,2 •, e" •+i/<2 and e? -, we sum over i and j, and we add them. We obtain, using a discrete inetegration by part:

thanks to Young's inequality and the regularity of u and v, one has:

and for all a G R, a > 0, thanks to Young's inequality and the regularity of p, we have:

Taking back [11] and [12] in [10], we obtain the result. For the second part, we use the regularity of (w,v,p) and of the mesh to replace the second order terms by some third order terms, changing all "h 2 " by "h 4 ". 3.2.2. Error estimate for the pressure Lemma 3.2

Under the assumptions 1, 2 and 3, and with the notation 2, there exists Ci depending only on (u, v,p) and fi such that:

Ifu£C3(Cl),v£

C 3 (fJ), p 6 C 2 (O) and if the mesh is uniform, then:

To prove this lemma, we use the following theorem proved by Shin and Strikwerda [SHI 97]. Theorem 3.2

Under the assumption 2, there exists C/ depending only of ft such that for all q e X(T), there exists (u,v) G X(T U ) x ~X(TV] satisfying:

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Applying this theorem to e?r we obtain a "velocity" (u, v) € X(T U ) x %(T V ). Then we replace ( u i + i / 2 , j ^ i , j + i / 2 j P i , j ) m the first and the second equations of the scheme by (e"+1/,2>J., e^-+1/2, ef^-), we multiply each respectively by u i+ i/2,j and by ^-+1/2, and we sum each equation on i and j and we add both to obtain:

with, thanks to the Cauchy-Schwarz inequality:

where C\ only depends on (u,v,p) and £1. Hence from [16],[17], [18] and [15] and thanks to Theorem 3.2, there exists C > 0 such as:

so by Lemma 3.1, with .

the result. For the second part of the

i.e. with 2

4

lemma, we replace again all "/i " by "/i " in the preceding reasoning. 3.2.3. Proof of theorem 3.1 The error estimate for the pressure is given by Lemma 3.2. For the velocity, we take back the value of a and C^ in the estimate [8], and we obtain:

Therefore the theorem is proved with C = max(C2, 64).

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References [CHO 98] CHOU S.H. AND KWAK D.Y., A Covolume method Based on Rotated Bilinears for the Generalized Stokes Problem. SIAM J. Numer. Anal, Vol. 35, N° 2, p. 494-507, April 1998. [EGH 99] EYMARD R., GALLOUET T. AND HERBIN R., Convergence of finite volume schemes for semilinear convection diffusion equations. Numer. Math., Vol. 82, p. 91-116, 1999. [EGH 97] EYMARD R., GALLOUET T. AND HEREIN R., Finite Volume Methods. Preprint N° 97-19 LATP, Aix-Marseille 1, to appear in Handbook of Numerical Analysis, P.O. Ciarlet, J.L. Lions eds. [FAR 97] FARHLOUL M. AND FORTIN M., A New Mixed Finite Element for the Stokes and Elasticity Problems. SIAM J. Numer. Anal, Vol. 30, N° 4, p. 971-990, August 1997. [HAN 98] HAN H. AND Wu X., A New Mixed Finite Element Formulation and the MAC Method for the Stokes Equations. SIAM J. Numer. Anal, Vol. 35, N°. 2, p. 560-571, April 1998. [HAR 65] HARLOW F.H. AND WELSH J.E., Numerical calculation of timedependant viscous incompressible flow of fluid with free surfaces. Phys. fluids, Vol. 8, p. 2181-2189, 1965. [NIC 92]

NiCOLAlDES R.A., Analysis and Convergence of the MAC Scheme I. The Linear Problem. SIAM J. Numer. Anal, Vol. 29, p. 15791591, 1992.

[NIC 97]

NiCOLAlDES R.A. AND Wu X., Covolume Solutions of ThreeDimensional Div-Curl Equations. SIAM J. Numer. Anal, Vol. 34, N° 6, p. 2195-2203, December 1997.

[PEY 83] PEYRET R. AND TAYLOR T.D., Computational Methods for Fluid Flow. Springer- Verlag, New York, 1983. [POR 78] PORSHING T. A., Error Estimates for MAC-Like Approximations to the Linear Navier-Stokes Equations. Numer. Math., 29, p. 291-306, 1978. [SHI 96]

SHIN D. AND STRIKWERDA J.C., Convergence Estimates for Finite Difference Approximations of the Stokes Equations. J. Aust. Math. Soc. Ser B, 38, p. 274-290, 1996.

[SHI 97]

SHIN D. AND STRIKWERDA J.C., Inf-sup Conditions for Finite Difference Approximations of the Stokes Equations. J. Austral. Math. Soc. Ser. B, Vol. 39, p. 121-134, 1997.

Convergence Rate of the Finite Volume Timeexplicit Upwind Scheme for the Maxwell System on a Bounded domain

Yves Coudiere INSA, complexe scientifique de Rangueil, 31077 Toulouse Cedex 4, France P. Villedieu ONERA Toulouse, 2 Avenue Edouard Belin, 31055 Toulouse Cedex, France

ABSTRACT : We derive an O(h1/2) error estimate for the upwind, explicit in time, finite volume scheme for Friedrichs systems. Explicit schemes in that case can not be seen as standart time-space finite element ones. Our demonstration is the generalisation on bounded domains of the ideas of Vila and Villedieu for the Cauchy problem in Md. It is applied to the case of Maxwell's equations. Key Words: hyperbolic system, Maxwell's equations, finite volumes, error estimates.

1. Introduction We are interested in the approximation by finite volume means of the Friedrichs systems of the form

d

where Ai are some symmetric matrices,An = 2_.^-ini (n being the unit nor»=i mal outward to £)). (1) belongs to the class of hyperbolic systems first introduced by Friedrichs [FRI 58]. For M such that ker (An — M) is maximal positive, it have a unique solution in L2(]0, T[x£}) [RAU 85].

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We shall treat, as an example, the case of the two dimensional Maxwell's equations in Transverse Magnetic Mode. Our aim is to derive an O(/i 1//2 ) error estimate for the upwind, explicit in time, finite volume scheme for (1), under very general assumptions on the mesh and minimal regularity hypothesis on the continuous solution u. The discontinuous Galerkin method, applied to the approximation of solutions of (1) is a space-time finite element method, which can be interpreted as a finite volume method, but with an implicit in time discretisation [JOH 87]. However, finite volume schemes with time-explicit discretisation can not be interpreted in general as discontinuous space-time finite element schemes associated to a coercive bilinear form. Derivation of error estimates of order /i 1 / 2 for such schemes in the case of the Cauchy problem on Md was first obtained by Vila and Villedieu [VIL 97, VIL 99], using a new technique of demonstration. On bounded domains, the additional difficulty is the discretisation of the flux Ann on the boundary of Q. Here, we propose a general form for the numerical flux on d£l, which guarantee the consistency and the stability of the corresponding scheme. We also prove an error estimate of order /z 1 / 2 (as for time implicit schemes). In section 2, we present the Maxwell's equations under the form (1). The scheme is defined in section 3. The actual results are stated in section 4. Sketches of the proofs can be found in section 5.

2. The Maxwell system As an example, we consider the bidimensional Maxwell system in Transverse Magnetic mode (TM waves). It may be written as a Friedrichs system of the form (1) in Q C M2, with

The unknown is u = [ E Hx Hy ] (Electric an Magnetic Fields; supposing that c = 1 is the light speed). We shall suppose that fi is the bounded domain between an obstacle and an outer boundary. Classical boundary conditions are - on the obstacle (d£l\), the metallic condition: E = 0,

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127

- on the outer boundary (dO,2), the linearized absorbing boundary condition of Silver Muller: H A next = E. These conditions may be stated like in (1) by taking

where the outward normal to <9Q is n ex t = [ ® P ] •

3. The numerical scheme

Let Th be a mesh of Q, composed of polyhedral cells K. In order to avoid any local degeneracy of the mesh, we assume that there exists some positive constants a and b such that

m(K), m(dK), diam(A') denote, respectively, the measure in Md of the cell K, the measure in M d - 1 of the boundary of K, the diameter of K. Let At be a time step, and tn = nAt. We shall approximate the solution of (1) by a piecewise constant function Vh such that

The values of i>/j are calculated according to the following scheme:

g^e are some numerical fluxes, defined below; B+ and B denote, respectively, the positive and the negative parts of a symmetric matrix B.

3.1 The interior numerical flux Let «5£ be the set of the interfaces interior to Q. We take the natural upwind scheme on such interfaces:

3.2 The numerical flux on the boundary

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Finite volumes for complex applications

Let dSh be the set of the edges of the boundary 5Q. We propose to take the following general form for the numerical flux on the boundary:

Since An = - (An + M) + - (An — M) and (An - M) u = 0 on the boundary, Zi Zt a natural choice is -yeQe = 0- But it yields an non stable scheme in general. • Qe is a stabilization term that does not modify the consistency of the flux. We shall take: Qe is the perpendicular projection on ker (Ane — Me) . • 7e > 0 is a parameter measuring the importance of the stabilization term. It shall have an effect on the CFL condition.

3.3 Case of the Maxwell system For the maxwell system given by the matrices (2), and the boundary conditions (3), a easy calculation shows that, on a boundary edge e: Metallic boundary condition:

• Absorbing boundary condition:

We point out that classical boundary numerical fluxes fall in this class for some particular choices of 7 [PIP 99]. • Metallic boundary condition: with the mirror state technique, the flux is given by taking (5) with vr^e = [ —E Hx Hy ] (if the interior state is vie = [ E Hx Hy f):

which is exactly the previous one, for 7 =

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• Absorbing boundary condition: the flux is given by taking v1ffe (no incoming waves):

129

— 0 in (5)

which is equivalent to the formulation above, only for the limit case 7 = 0.

4. Main Results Under some regularity assumptions on the MO, the Ai and £7, (1) admits a unique solution u <E V = C°([0, T\, (H1^))™) n C^QO, T], (£ 2 (Q)) m )

[RAU85]. Theorem 1 (Comparison) For T > 0, suppose that v € £ 2 (]0, T[xQ) 25 suc/i that • there exists ^ £ V (the dual space of V) such that Vip € V,

• there exists a measure v such that

then with Theorem 2 (Convergence Rates) Under the following CFL conditions: • on the interior interfaces,

• on the boundary edges,

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Finite volumes for complex applications

the approximate solution v^ given by the finite volumes scheme (4) converges to the exact solution u of (1), and we have

5. Proofs

5.1 The Theorem of Comparison An easy calculation yields, for any £ C1([0, T] x fi) such that (j>(t, x] > 0 on]0, T[xfi,

The theorem 1 is obtained by taking (j)(t, x) = T — t (dt

0 which tends to zero as size(T) -> 0 such that 0 which tends to zero as size(T) —> 0 such that

where (interior case), (boundary case) Calculating

we find that

• Interior case:

using the CFL condition in a classical way.

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131

• Boundary case:

where

Summing over the e £ dK, and coming back to the conservative form, we find that the following inequality should be satisfied on the boundary of O:

It is the case under the CFL condition of theorem 2 (and because Qe is the projection on ker (Ane — Me) ). Remark that, for the absorbing boundary condition, the form of Me enables to conclude even if 7 e =0 (ie for the classical numerical flux). Finally, summing over the cells K £ Th and the time steps n = 0 .. . N wehave

5.3 The Errors of Consistency We start by stating the expressions of the Hh and Vh that will be used ii the theorem of comparison (with some obvious notations):

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Finite volumes for complex applications

(0g and ^g are average values on ]tn, tn+1[xe, and #-, T/>_R- are average values on A"). The final estimates are deduced from the inequality of comparison of theorem 1. We have to calculate (i/h, <(>) — 2{/j/ l , ufi) for <j)(t, x) = T — t. (7) and (11) are easily estimated if u0 G H1^}. (8), (9) and (12), (13) are combined and bounded by the L2 norms of Vh and its discrete derivatives, using Cauchy Shwarz inequalities and a suitable Young inequality. (10) is bounded as wanted because Qeu™ = 0 (u is exact solution). See [COU 99] for the details. [COU 99]

Y. Coudiere. Analyse de schemas volumes finis sur maillages non structures pour des problemes lineaires hyperboliques et elliptiques. PhD thesis, Universite Paul Sabatier, 12 Janvier 1999.

[FRI 58]

K.O. Friedrichs. Symmetric positive linear differential equations. Comm. Pure and Applied Math., 1958.

[JOH 87]

C. Johnson. Numerical solution of partial differential equations by the finite element method. Cambridge University Press, 1987.

[PIP 99]

S. Piperno. L 2 -stability of finite volumes schemes for the Maxwell system in two and three dimensions on arbitrary unstructured meshes, submitted, 1999.

[RAU 85]

J. Rauch. Symmetric positive systems with boundary characteristic of constant multiplicity. Trans, of the AMS, 1985.

[VIL 97]

J.P. Vila and P. Villedieu. Convergence de la methode des volumes finis pour les systemes de Friedrichs. C.R. Acad. Sci. Pans, 1997.

[VIL 99]

J.P. Vila and P. Villedieu. Convergence of the Time-Explicit Finite Volume Scheme for Linear Symmetric systems, submitted to SIAM Journal of Numerical Analysis, 1999.

Flux vector splitting and stationary contact discontinuity Frangois Dubois Conservatoire National des Arts et Metiers 15 rue Marat, F-78 210 Saint Cyr PEcole, France. [email protected] Abstract In order to capture in a stable and accurate way a boundary layer with an upwind finite volume scheme, the numerical analysis of a stationary contact discontinuity problem shows that under natural symmetry hypotheses, a flux splitting generates a numerical viscosity proportional to the difference of densities whereas the numerical viscosity is null for a flux difference splitting approximating all the waves of the Riemann problem.

1) Introduction. • We study the Euler equations of gas dynamics in one space dimension. They take the form of an hyperbolic system of conservation laws between dW d state W and physical flux f(W) : 4- — f(W) = 0 where state ot ox W = W(t,x) € IR3 represents a volumic density of mass, momentum and energy : W = (p, pu, pE = pe + —pu2] . The algebraic form of physical flux f ( W ] e H 3 is given by

(1)

^

f(W) = (pu,pu2 + p, puE + puf ;

it uses pressure, which is a function of density p and of internal energy e parameterized for a polytropic perfect gas by ratio 7>1 of specific heats : P = (7 - 1) pe • • In order to approach numerically the solutions of the gas dynamics equations, we introduce the so-called finite volume method ; space is discretized with a grid j Ax (j € 7L] and time by multiples n At (n e IN) of time step At. We search an approximate value W? of the field W(«,«) at particular vertex j Ax and discrete time level n At thanks to the family of numerical fluxes f"+i/£ (j e Z, n e IN) (see e.g. Harten, Lax and Van Leer [HLV83]) : ^(W7 +1 - W?) + ±(f£$ - y^/22) = 0. In this note, we restrict ourselves to a two-point numerical flux function that is explicit and first order accurate in space and time, e.g. of the form : fn+i/2 ~ $(Wj*) Wj+i)-

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• We distinguish between two types of numerical flux functions depending of two arguments : on one side, exact or approximate solutions of the Riemann problem ("flux difference splitting") between states WJ1 and W^+l (see e.g. Godlewski-Raviart [GR96] for mathematical and numerical context) with the numerical fluxes proposed by Godunov [Go59], Roe [RoeSl] and Osher [Os81] and on the other side flux decompositions ("flux vector splitting"). A flux vector spliting, with Sanders-Prendergast [SP74], Van Leer [VL82], Bourdel, Delorme and Mazet [BDM89] or Perthame [Pe91] among others, suppose that the physical flux function R3 3 W \—> /(W) € IR3 explicited in (1) has been written under the form (2) f ( W ] = f+(W) + f-(W] with a set of constraints on functions / + (») and /"(•) detailed for example in the book of Godlewski and Raviart [GR96]. For modelling upwinding, the numerical flux admits the following very simple form : (3) fc(W<, Wr] = f+(Wt} + f-(Wr). • In the context of a stationary aerodynamics problem, Van Leer, Thomas, Roe and Newsome [VTRN87] compare the Van Leer flux vector splitting [VL82] and the Roe scheme [RoeSl] that uses a Riemann problem for a linearized equation. They show that in order to give a correct prediction of skin friction and heat flux on the boundary with a relatively course grid, it is possible with the Roe flux difference splitting while it is not with the Van Leer flux vector splitting of the type (3) for convective part of fluid flow. Their conclusion is to reject flux vector splitting methodology if the objective is to predict more than the simple pressure field. • In fact, the problem occurs in the boundary layer. Along the direction x normal to the boundary, normal velocity u is very small. Then it is natural to study the evolution of a flux vector splitting (3) for the very simple model of a stationary contact discontinuity, i.e. a boundary layer with infinitesimal thickness. It is a particular problem of decomposition of discontinuity where the given states Wi and Wr define on one hand a velocity field identically null composed by u/ for x < 0 and by ur for x > 0 : (4) m = ur = 0, and on the other hand a pressure field denoted respectively by pi for x < 0 and pr for x > 0 without discontinuity : (5)

pi = pr

=

p.

The physical solution of such a stationary contact discontinuity does not depend on time : density jump is maintained at the interface x = 0 as long as time variable is increasing and it is the addition of viscous term or of geometrical perturbations like in Kelvin-Helmholtz instability that modify the interface, which is crucial for a correct capture of boundary layers and shear instabilities. • In this note, we prove that in a general way if a flux vector splitting satisfies very natural hypotheses of left-right invariance (section 2), then the associated scheme for gas dynamics contains a numerical viscosity essentially proportional to the jump of density, then of the order zero relatively to space step (section 3).

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135

2) Left-right invariance. • We consider transformation a of state W obtained by changing the sign of velocity : Taking into account the particular algebraic form of state W and relation (1), we observe that when we change the sign of velocity, we change the sign of mass flux and of energy flux but we do not the sign of momentum flux. • Because changing the sign of velocity is equivalent to changing the sign of space direction x, it is usefull to introduce the normal unitary vector n to this direction (n 6 { — 1,1}), to set g(n, W} = ( p ( u » n ) , (pu2 + p)n, (pE + p)(u» n)) and also an — —n. If we change both signs of velocity and of space direction, the mass and energy fluxes remain unchanged but the sign of momentum flux is changed. We have in consequence : • Natural extension of this left-right invariance property to the numerical flux can be formalized by setting :

The left-right invariance for numerical flux consists to remark that if we exchange both left and right states, the sign of their velocity and the normal direction, we only change the sign of momentum flux. • Definition 1. Left-right invariance property. The numerical flux function (Wj, Wr) i—> $(Wj, Wr) satisfies the left-right invariance property if the function \I>(», •, •) defined in (9) and operator a defined at relation (6) and by an — — n satisfy the condition • We remark that consistency condition $(W, W} = f(W) can be translated for the pair ( ^ , g ) by the relation fy(W, n, W} = g(n, W) and in this particular case, relation (8) shows that ty(aW, crn, aW) — a^/(W, n, W) = g(crn, aW] - crg(n, W} — 0. This remark establishes a particular case of relation (10) when Wi = WT = W. • Proposition 1. Left-right invariance of a flux vector splitting. A flux vector splitting (2) associated with a numerical flux function (3) satisfies the left-right invariance property if and only if we have Proof of Proposition 1. • We introduce the representation (3) inside relation (10) when n — +1 :

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Finite volumes for complex applications

for each pair (W/, Wr). If we make the choice of two independent states W/ and Wr the preceding relation states clearly the relation (11) and when we explicit the action of the operator a on a vector (see relation (6)), we obtain the detail of the algebra for each component, i.e. relation (11). • On the other hand, if condition (11) is satisfied, then the relation (10) is correct for n = 1 ; it remains true for n = — 1 because the left member is an odd function of variable n due to relation (9). In consequence the proposition is established. D • Proposition 2. Particular case of classical flux vector splittings. Van Leer flux, Sanders Prendergast flux and Boltzmann schemes satisfy the left-right invariance property. Proof of Proposition 2. • Van Leer flux satisfy the following relation f+(W]

U

= f(W)

if M = — > c

I and f~(W) = f(W) if M < -1. If the Mach number M of state W is greater or equal to 1, then the Mach number of state aW is lower or equal to -1 ; then f+(aW} = 0 = -af~(W) and relation (11) is established in this case. If on the contrary M < — 1, then f+(&W) = f(ffW) and /~ (W) = f(W) and relation (11) is in this case a simple re-writing of relation (7). • When M | < 1, Van Leer flux vector splitting satisfies the relation

and relation (11) is clear. Consistency condition (2) is not obvious ; the proof is an algebraic calculus introduced in the original work [VL82]. •

In the case of a Boltzmann scheme, we write the state W under the particu-

lar form W = I

J-oo

x\

7=—) (l> vi ~ Iv |2) dv where x(*) is a positive

^ vT

'

2

function that defines the numerical scheme and T is the temperature. The flux +

°°

(\V-U\\

V

t

X\ 7=—) (v, v |2, - | v |2) dv. 2 / -oo ^ vT ' To take into account the fact that flux /+ represents the action of all the particles going from the left to the right, we set is simply evaluated by

and in an analogous way, due to (2) :

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137

(14) After having made the change of variable v i—> —v inside the integral (13), we deduce

and relation

(11) est clear. • We consider now the case of Sanders and Prendergast splitting ; we just replace function \ of relation (13) by a linear combination of Dirac measures at particular points u — c, u, + c in velocity space, where c is the sound waves celerity, in order to satisfy the following relations

Then total energy pE is decomposed under the form and the particular algebraic form that controls the measure

allows to deduce

(16) In an way analogous to the other Boltzmann schemes, the flux vector splitting results from an integration on the interval ]0, +oo[ to evaluate f+ and on the opposite interval ] — oo, 0[ for f-. Even parity of f^ is a consequence of parity of energy decomposition in (15) whereas odd parity of ff and /* is a consequence of imparity of impulse in (15) and of relation (16). This result establishes relation (11) for Boltzmann schemes and Proposition 2 is proven. Q 3) Numerical viscosity on a stationary contact. • Definition 2. Numerical viscosity. Numerical viscosity l^(Wi, Wr) of a two-point numerical scheme of the type is defined by the relation

• Proposition 3. Numerical viscosity of a flux vector splitting. Let <£(•, •) be a flux vector splitting of the type (3). Then numerical viscosity V(Wi, Wr) satisfies the relation Proof of Proposition 3. • It results from the following calculus : due to

and relation (18) is established.

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Finite volumes for complex applications

• Proposition 4. Stationary contact discontinuity. Let W be a state with a velocity equal to zero. It satisfies in particular (19) aW = W. Then if the flux vector splitting (•, •) defined in (2)-(3) satisfies the left-right invariance property, there exists two functions IR and in order to satisfy

Moreover, for a stationary contact discontinuity (4)-(5), the numerical viscosity satisfies •

In the case of Van Leer flux vector splitting, relations (12) show

and for a Boltzmann scheme, we have, taking into account (13) and (14),

Proof of Proposition 4. • Relations (19) and (11) show that Joined with relation (2), we have from relation (22) f(W) if aW = W and relation (20) is established for function /+(•)• The entire relation (20) is a direct consequence of (22). The detail of the computation of numerical viscosity is a consequence of the relations (18) and (20). Q • Proposition 5. Residual numerical viscosity. If one of the functions //(», •) and e(», •) explicitly depends on density, i.e. i f w eh

a

v

e

t

h

e

n t h e numerical viscosity

of a flux vector splitting scheme is not infinitesimal for a stationary contact discontinuity, whatever be the size of the mesh. Proof of Proposition 5 is an immediate consequence of Proposition 4 and in particular of the relation (21). D • Proposition 6. Approximate Riemann solver. Let be one of the three exact or approximate Riemann solvers proposed by Godunov [Go59] (exact solution of the Riemann problem), Osher [Os81] (approximate solver containing only rarefaction waves or contact discontinuity) and Roe [RoeSl] (approximate solver containing only contact discontinuities). Then numerical viscosity V(W/, Wr) of such a numerical scheme is null if given states Wi and Wr satisfy the particular conditions (4)-(5) of a stationary contact discontinuity.

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139

Proof of Proposition 6.

• In the case of Godimov and Osher schemes, conditions (4)-(5) state that the solution of the Riemann problem is effectively only composed by a stationary contact discontinuity. Due to Rankine-Hugoniot relations for a stationary discontinuity, physical fluxes of the two states Wi and Wr are equal and we have /* = (0, p, 0)* = /(Wi) = /(W r ). Taking into account relation (7) in Definition 2, the result is established in this case. • If we use the Roe flux, we compute in a first step [RoeSl] intermediate velocity u* of a mean state :

due to relation

(4) and we evaluate also total enthalpy of this mean state before the calculus of Roe matrix A* that satisfies, taking into account previous expression of The difference (Wr - Wi) is an eigenvector of matrix A* relatively to the eigenvalue u* = 0 due to the expression of intermediate velocity u*. In consequence, when we decompose discontinuity Wr — W/ on the basis of eigenvectors for matrix A*, we observe that this difference is non null only for the linearly degenerated wave, i.e. on the contact discontinuity itself. Conclusion is then exactly the one done previously for Godunov and Osher fluxes. D 4) Conclusion. • In order to capture numerically a boundary layer with a finite volume scheme, numerical analysis of the problem of stationary contact discontinuity shows that classical flux vector splitting schemes satisfying the left-right invariance property generates a numerical viscosity of order one relatively to the jump of densites whereas it is not the case if we use an exact or approximate decomposition of the Riemann problem. This fact founded on very simple algebra shows that Van Leer at al conclusion can be extended to all flux vector splittings referenced in this note : flux vector splitting satisfying the left-right invariance is incompatible with viscous computations. • The previous remark conducted us during the time of development of the Navier Stokes solver NsSgr to include the Osher flux whereas the initial choice was the Sanders and Prendergast flux vector splitting. This choice has been performing, even for the resolution of the Euler equations of gas dynamics in the particular case of capturing shear stationary waves (see [DM92]). In an analogous way, the parabolized version FluSpns (Chaput et al [Ch91]) of FluSc computer software has required the introduction of the Osher flux decomposition in order to simulate flows with a precise evaluation of viscous effects. 5)

References.

[BDM89] F. Bourdel, P. Delorme, P. Mazet, Proceedings of the 2th International Conference on Nonlinear Hyperbolic Problems, Notes on Numerical Fluid Mechanics, vol. 24, p. 31-42, Vieweg, 1989. [Ch91] E. Chaput, F. Dubois, G. Moules, D. Lemaire, J.L. Vaudescal. A Three Dimensional Thin Layer and Parabolized Navier-Stokes Solver Using the MUSCL Upwind Scheme, AIAA Paper n° 91-0728, January 1991.

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[DM92] F. Dubois, O. Michaux. Solution of the Euler Equations Around a Double Ellipsoidal Shape Using Unstructured Meshes and Including Real Gas Effects, Workshop on Hypersonic Flows for Reentry Problems (Desideri-Glowinski-Periaux Editors), Springer Verlag, vol. 2, p. 358373, 1992. [Go59] S.K. Godunov. A Finite Difference Method for the Numerical Computation of Discontinuous Solutions for the Equations of Fluid Dynamics, Mat. Sbornik, vol. 47, p. 271-290, 1959. [GR96] E. Godlewski, P.A. Raviart. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118, Springer, New York, 1996. [HLV83] A. Harten, P.D. Lax, B. Van Leer. On Upstream Differencing and Godunov-type Schemes for Hyperbolic Conservation Laws, SIAM Review, vol.25, n° 1, p. 35-61, January 1983. [Os81] S. Osher. Numerical Solution of Singular Perturbation Problems and Hyperbolic Systems of Conservation Laws, in Mathematical Studies n°47 (Axelsson-Franck-Van der Sluis Eds.), p. 179-205, North Holland, Amsterdam, 1981. [Pe91] B. Perthame. Second Order Boltzmann Schemes for Compressible Euler Equations in One and Two Space Variables, SIAM Journal of Numerical Analysis, vol.29, p. 1-19, 1991. [RoeSl] P. Roe. Riemann Solvers, Parameter Vectors, and Difference Schemes, Journal of Computational Physics, vol.43, p. 357-372, 1981. [SP74] R.H. Sanders, K.H. Prendergast. The Possible Relation of the 3kiloparsec Arm to Explosions in the Galactic Nucleus, The Astrophysical Journal, vol. 188, p. 489-500, march 1974. [VL82] B. Van Leer. Flux-Vector Splitting for the Euler Equations, Proceedings of the ICNMFD Conference, Aachen 1982, Lectures Notes in Physics, vol. 170, p. 507-512, Springer Verlag, 1982. [VTRN87] B. Van Leer, J.L. Thomas, P. Roe, R.W. Newsome. A Comparison of Numerical Flux Formulas for the Euler and Navier Stokes Equations, AIAA Paper n° 87-1104, AIAA 8th CFD Conference, 1987.

Analysis of a Finite Volume Solver for Maxwell's Equations

Fredrik Edelvik Department of Scientific Computing, Uppsala University Box 120, S-751 04 Uppsala, Sweden [email protected]

ABSTRACT A finite volume solver for Maxwell's equations is analyzed. The solver shows excellent dispersion characteristics on three different uniform triangular grids. Long term stability is achieved for general unstructured grids using a third order staggered Adams-Bashforth scheme for the time discretization. The solver has been hybridized with a finite difference solver and the resulting hybrid solver is shown to be second order accurate for a 2D cylinder scattering case. Key Words: Finite volumes, Maxwell's equations, Dispersion, Hybrid grid

1. Introduction The most popular numerical method for solving electromagnetic problems is the Finite-Difference Time-Domain (FDTD) method [TAF 95]. It is normally used on a Cartesian grid and the variables are staggered both in time and in space. The main disadvantage with FDTD is its inability to model complex shaped geometries producing staircasing errors. One way of solving this problem, but still taking advantage of the efficiency of the FDTD method, is to use a hybrid grid solver. A Finite-Volume Time-Domain (FVTD) solver on an unstructured grid is used in the near vicinity of the object and this solver is coupled to an FDTD solver on a structured Cartesian grid that is used in the outer region. This approach was proposed by Riley et al. [RIL 97] and it is also used in the Swedish Computational Electromagnetics project GEMS (General Electromagnetic Solvers). GEMS is a collaborative research and code development project between Swedish industry and academia. This paper con-

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Finite volumes for complex applications

centrates on the finite volume solver. The solver is described in Section 2 and its dispersion characteristics and stability properties are analyzed in Sections 3 and 4, respectively. In Section 5 some numerical results obtained with the hybrid solver are presented and the last Section summarizes the paper and some conclusions are drawn for future work. 2. The Finite Volume Solver The finite volume (FV) solver has so far been implemented in 2D for solving the transverse magnetic (TM) Maxwell's equations,

for a linear, isotropic, non-conducting and non-dispersive material. Here Hx and Hy are the x- and y-component of the magnetic field, Ez is the z-component of the electric field, p, is the magnetic permeability and e is the electric permittivity. The basis for the FV solver is the following integral form (in 2D) of Maxwell's equations

where A is an arbitrary area and the line integral is taken along the path F that encloses A and n is a unit normal. In 2D the computational domain is discretized using a staggered unstructured grid, consisting of a primary grid of triangles and a dual grid (cf. Fig. 1). For TM mode, Hx and Hy are stored at the nodes, whereas Ez is stored at the barycenters of the primary cells. By joining barycenters of neighboring cells the dual grid is constructed during the pre-processing stage. The area integrals are evaluated by the assumption that the electric and the magnetic field components remain constant over a primary and a dual cell, respectively. Simplifying the two integrands in the line integrals implies

where H and Ez denote average fields and the sums are taken along the edges with unit vector t of the respective cells. A^ and Ap are the areas of the dual

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143

and primary cells. The evaluation of the first line integral is performed by the assumption that the electric field is piecewise constant along each dual edge. Following Riley et al. [RIL 97], the second line integral is evaluated according to

where HI is the FDTD component in the direction n/ orthogonal to the dual edge crossing the primary edge t, H; and H& are the magnetic field in the nodes of the edge t. HI is incorporated in the edge projected field value to guarantee that the divergence is preserved on a local cell level. This is very important, since not fulfilling the two divergence relations of Maxwell's equations could cause spurious modes that would destroy the solution. The time integration method that we use is a third order staggered AdamsBashforth (ABS3) scheme proposed by Fornberg et al. [FOR 99] for scalar wave equations. For the TM Maxwell's equations the scheme takes the form

where A and B are operators taking care of the space discretization. 3. Dispersion Analysis on Triangular Grids

A frequently used technique to characterize the errors of a numerical scheme is to use Fourier analysis. Assuming that the fields are periodic in space and neglecting boundary conditions, we make the following ansatz for the three unknown field components

where kx and ky are the x- and y-components of the numerical wave vector, respectively. The numerical wavenumber k will in general differ from the physical wavenumber k defined by k — ui/c, where u> is the angular frequency and c is the speed of light, k depends on how well the wave is resolved in the grid, the type of grid and the angle of propagation in the grid. The difference between k and k gives rise to numerical phase and group velocities that depart from the exact values, which cause numerical errors that accumulate with distance. In figure 1 the different uniform grids used in the analysis are shown. Note that the normals and edge vectors align in the equilateral and diamond grid. Thus, only the FDTD correction HI survives expression (6), so for those two grids we do not have to take the magnetic field in the nodes into account. For

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Finite volumes for complex applications

the one directional grid, however, we have to consider both the magnetic field at the nodes and the FDTD correction. Since the approximation of the dispersion relation is in the midpoint of the respective building blocks shown in figure 1, in the analysis, the Ez components are interpolated from the barycenters to the midpoint of the building blocks.

Figure 1: The different uniform triangular grids used in the analysis, the equilateral grid, the one directional grid and the diamond grid. The respective dual grids are indicated by dashed lines. Let us first look at the errors introduced by the space discretization for the equilateral grid. Substituting (9) into (8) and assuming that no error is introduced by the time discretization results in

Similarly, from (7) we have for the three edges (see Fig. 1)

Inserting (11)-(13) into (10) leads to the numerical dispersion relation

Letting A —> 0 we obtain

i.e. the dispersion relation is approximated to second order accuracy. The same analysis on a square grid with edge length A, where the solver is identical to the

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FDTD scheme, gives an error constant equal to ^ [TAP 95]. The dispersion relation is approximated to second order accuracy also for the grids with right triangles, but since these two expressions are somewhat lengthy they will be omitted here. The dispersion error characteristics can be presented in terms of the error in the phase speed of waves of different grid resolution and propagation angle. Let kx = kcos(a) and ky = ksin(a), where a is the angle of propagation relative to the x-axis. Substituting these two expressions into the dispersion relation (14) and solving the nonlinear equation for fc, results in a relation between the numerical wave speed, vp, and c, since vp/c — k/k. Figure 2 illustrates the numerical phase velocity as a function of propagation angle and grid resolution. The results obtained by the FV solver are in excellent agreement with the true wave speed for all three grids even for a moderate grid resolution of 10 points per wavelength. The dispersion characteristics are clearly better for triangular grids than for square grids.

Figure 2: Variation of numerical phase velocity for different grids as a function of grid resolution and propagation angle. Discretizing the time derivative with ABS3 we obtain the following fully discrete dispersion relation for the grid with equilateral triangles

The time step is chosen according to the stability limit for ABS3 (see next Section) as cAt = \/6A/7. As in the semi-discrete case we solve (16) for k. Since ABS3 is a third order accurate scheme, the errors from the space discretization will dominate and a plot of vp/c shows almost identical results as in figure 2. If we use the Leap-Frog scheme for the time-stepping we get a small improvement in the dispersion characteristics since the scheme is only second order accurate and thus the errors from the space and time discretization cancel to some extent.

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4. Stability Analysis of the FV Solver An important issue for explicit solvers across many disciplines is how to achieve long term stability without adding too much artificial dissipation. The stability region for ABS3 in the scalar case is given in figure 3.

Figure 3: Stability region for third order staggered Adams-Bashforth. The scheme is stable between ±12/7 along the imaginary axis compared to the Leap-Frog scheme, which is stable between ±2. That implies that we have to use a shorter time step for ABS3. However, the main disadvantage with Leap-Frog is that it is only stable on the imaginary axis and becomes unstable as soon as we have eigenvalues with a nonzero real part, which we are likely to have on unstructured grids and when boundaries come into play. Let us analyze how ABS3 behaves in our case. After some straightforward algebra (7) and (8) can be written in matrix form as

The plane wave ansatz

leads to the following generalized eigenvalue problem

where A,B and C are the matrices in (17), respectively. We begin by looking at the uniform triangular grids used in the analysis of the dispersion error. For the grid consisting of equilateral triangles we get the matrices A and B from (10)-(13). A stability condition for the Leap-Frog scheme on the three triangular grids is easily derived and equals

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where the first part in the right-hand side is the stability condition on Cartesian grids and miniength is equal to the shortest edge in the unstructured region. When we solve the generalized eigenvalue problem (18) on these grids we observe that stability is obtained if the time step suggested by (19) is decreased by a factor 6/7, which is exactly the relative difference in stability along the imaginary axis (see Fig. 3) between the two methods. To analyze the eigenvalues for a general unstructured grid including boundaries we can no longer use Fourier analysis. Instead, let

and after some straightforward rearrangements we are able to write (7) and (8) on matrix form as zn+l = P(A,B}zn. Analyzing the eigenvalues of the iteration matrix P, for the grid shown in figure 4, reveals that if we choose the time step for ABS3 in the same manner as above we get the eigenvalue spectrum shown in figure 4, where all eigenvalues are within the unit circle. If we use the Leap-Frog scheme with the same time step the largest eigenvalue is of the order 1.0003. Hence the Leap-Frog scheme is unstable even for a time step well within the stability limit along the imaginary axis.

Figure 4: Eigenvalues of the iteration matrix P using ABS3 for a PMC cylinder scattering case. Primary grid around the cylinder is shown to the right. The long term stability using ABS3 on general unstructured grids around a PMC and a PEC cylinder has also been verified by feeding the FV solver along one of the outer boundaries with a narrow square wave pulse for ten million time steps without any signs of instability. 5. Numerical Results

The FV solver has been hybridized with a finite difference solver and the resulting hybrid solver has been tested on two scattering cases, a PMC and a PEC cylinder. In both cases a plane Gaussian shaped wave is impinging on the cylinders. In figure 5 the results for different resolutions using the hybrid

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solver are compared to a well resolved FDTD solution. An analysis of the results reveals that the hybrid solver is second order accurate for both cases. The hybrid code has been verified to be long term stable in the same manner as the stand alone FV solver.

Figure 5: Errors for hybrid solutions of different the left and PEC cylinder to the right.

resolution. PMC cylinder to

6. Conclusions We have demonstrated that our FV solver has excellent dispersion characteristics on three different uniform triangular grids. A third order staggered Adams-Bashforth scheme is used instead of the normally used Leap-Frog scheme. Indications on how to choose the time step for ABS3 on general unstructured grids are given. The solver has been shown to be long term stable for two cylinder test cases. The implementation of the FV solver in 3D is ongoing and the hybrid technique will be used there as well. 7. Acknowledgment The author would like to thank Ulf Andersson and Gunnar Ledfelt at KTH, Stockholm, who participated in the hybridization of the code and performed the numerical tests in Section 5. 8. Bibliography [TAF 95]

TAFLOVE A., Computational Electrodynamics, The FiniteDifference Time-Domain Method, Artech House, Norwood, 1995.

[RIL 97]

RILEY, D.J. et al. , «VOLMAX:A Solid-Model-Based, Transient Volumetric Max well-Solver Using Hybrid Grids », IEEE Antennas Propagat. Magazine, 7V° 39, 1997, p. 20-33.

[FOR 99]

FORNBERG, B. et al. , «Staggered Time Integrators for Wave Equations », submitted to SIAM J. Sci. Comput.

A result of convergence and error estimate of an approximate gradient for elliptic problems

Robert Eymard1, Thierry Gallouet2, Raphaele Herbin2 l Ecole Nationale des Fonts et Chaussees, Marne-la-Vallee, France 2 Universite de Provence, Marseille, France.

ABSTRACT Using a classical finite volume piecewise constant approximation of the solution of an elliptic problem in a domain ft, we build here an approximate gradient of the solution. It is then shown that this approximate gradient converges in Hdiv(ty- An error estimate is given when the solution to the continuous problem belongs to H2(£l). Key Words: elliptic equations, finite volumes, gradient, convergence, error estimate.

1. Introduction As a paradigm of elliptic problems, we consider the Laplace equation

with Dirichlet boundary condition:

where we make the following assumption. Assumption 1 1. ft is an open bounded polygonal subset o/IR , d = 2 or 3,

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Here, and in the sequel, "polygonal" is used for both d = 2 and d = 3 (meaning polyhedral in the latter case). Note also that "a.e. on <9fT' is a.e. for the d — 1-dimensional Lebesgue measure on <9fl. Finite volume methods for Problem (1),(2) have been intoduced by many authors (see [EGH 99] and references therein), with some proofs of convergence and error estimates. Since the approximate solution constructed with a classical cell-centered finite volume scheme is piecewise constant, an approximation of the gradient of the solution may be seen to be more complex than with a finite element method. Indeed, the convergence of a reconstructed gradient has been shown in [CVV 97], for certain quadrangular meshes using a nine point scheme. It has also been shown on certain meshes by rewriting the finite volume scheme as a finite element scheme [ABM 95], [VAS 98], [AWY 97] or a Petrov-Galerkin scheme [DUB 97]. Here we show that one may construct an approximate gradient on all "admissible" meshes by using some mesh functions which generalize those used in mixed finite element theory (see e.g. [ROT 91]).

2. Approximate of u and Vu The following definition of admissible meshes for the finite volume scheme includes a large variety of meshes, such as, in two space dimensions, triangular meshes (with a four-points scheme), cell-centered or vertex-centered rectangular meshes (with a five-points schemes) and Voronoi meshes. Definition 1 (Admissible meshes) Let 17 be an open bounded polygonal subset oflRd, d = 2, or 3. An admissible finite volume mesh of ft, denoted by T, is given by a family of "control volumes", which are open polygonal convex subsets of fi (with positive measure), a family of subsets of fi contained in hyperplanes of TRd, denoted by £ (these are the edges (2D) or sides (3D) of the control volumes), with strictly positive (d — I)-dimensional measure, and a family of points of fi denoted by P satisfying the following properties (in fact, we shall denote, somewhat incorrectly, by T the family of control volumes): (i) The closure of the union of all the control volumes is fi; (ii)

For any K € T, there exists a subset SK of £ such that dK = K\K \Jff££Ka. Let E = UK<ET£K-

=

(Hi) For any (K, L) £ T2 with K ^ L, either the (d—I)-dimensional Lebesgue measure of K fl L is 0 or K n L — cf for some a € 8, which will then be denoted by K\L. (iv)

The family P = (XK}KET ™ such that XK € K (for all K G T) and, if a — K\L, it is assumed that the straight line T>K,L going through XK and XL is orthogonal to K\L.

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In the sequel, the following notations are used. The mesh size is defined by: size(T) = sup{diam(/C), K G T}. For any K e T one? cr e £, m(/0 is t/ie d-dimensional Lebesgue measure of K (i.e. area if d = 2, volume if d — 3) and m(cr) the (d — \)-dimensional measure of a. The set of interior (resp. boundary) edges is denoted by £jnt (resp. £ext), that is £[nt = {cr 6 £; a <£ dft} (resp.

A proof of the convergence in L 2 (fi) of the approximate solution uj- to the unique variational solution u G HQ(£I) of Problem (1), (2) and error estimates are given for example in [EGH 99]. We now build an approximate of Vw, the gradient of the solution of the continuous problem. To this purpose, we introduce, for K G T and a € £K the solution

The functions V0x,

by

3. Convergence theorem The following convergence theorem states the convergence of the approximate solution UT to the continuous one u in L 2 (f)) and the convergence of Gjto Vw in Hd\v(ty-

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Theorem 1 Under Assumption 1, let T be an admissible mesh (in the sense of Definition 1) . Let uj- be defined by (3)-(5) and let u € HQ(Q) be the unique variational solution of Problem (1), (2). Then HI—> u in L2(ft) as size(T) tends to 0. Furthermore, assume that there exist two fixed values £ > 0 and M such that the inequalities dK,ff > (diam(K} and M > card(£K} hold for any control volume K € T and for any a 6 EK • Let Gj- be the approximate gradient defined by equations (6)-(7). Then Gj- converges in H^\v(^l] to the gradient of the unique variational solution u G HQ (fi) of Problem (1), (2) as size(T) ->• 0. Sketch of the proof (the complete proof of this theorem is given in [EGH]). First note that for all K G T and for a.e. x 6 K,

and therefore divGr ->• —/ in L 2 (H) as size(T) tends to 0. Assuming only u € -ffo(ft), let e > 0 and (p € C£°(ft) such that

l|w-vll/n(n) 2 < £ •

Using the variational formulation of (6) in each K G T, one proves the existence of some Fi(M, £) > 0 and of some F2(fi,/,

With expressing the consistency of the fluxes, and following the steps of the I/2 (fi) error estimate proof, one then proves the existence of some F3 (fi, /,

Using the triangular inequality, one has

Choosing size(T) small enough such that F 2 (fi,/,^M,C,r)+F 1 (M,C)F 3 (n,/,^r)<£, equations (9)-(12) lead to

which shows that G-y > Vw in L 2 (fi) as size(T) ->• 0. Using (8), we get GT -)• Vu in F div (f)) as size(T) -)> 0.

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4. Error estimate Theorem 2 Under Assumption 1 let £ > 0 and M > 0 be given values and T be an admissible mesh (in the sense of Definition 1) such that the inequalities di<,ff > ^diam(K) and M > card(EK) hold for all control volume K G T and for all a 6 £K • Let uj- be given by equations (3)-(5). Then there exists c, only depending on £l, u and (,, such that

Furthermore, letGr be defined by equations (6)-(7). Assume that the unique variational solution, u, of Problem (1), (2) belongs to H2(£l). Then there exists C > 0 which only depends on H, u, £ and M such that

The proof of this theorem entirely follows the steps of the proof of Theorem 1, replacing (p by u 6 H2(fy (it is given in detail in [EGH]).

Bibliography

[ABM 95]

AGOUZAL, A.,BARANGER, J.,MAITRE, J.-F. andF. OUDIN (1995), Connection between finite volume and mixed finite element methods for a diffusion problem with non constant coefficients, with application to Convection Diffusion, EastWest Journal on Numerical Mathematics., 3, 4, 237-254.

[AWY97]

ARBOGAST, T., WHEELER, M.F. and YOTOV, I.(1997), Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SI AM J. Numer. Anal. 34, 2, 828-852.

[CVV 97]

COUDIERE, Y., VILA, J.P. and VILLEDIEU, P. (1997), Convergence Rate of a Finite Volume Scheme for a Two Dimensionnal Convection Diffusion Problem, accepted for publication in M2AN.

[DUB 97]

DUBOIS, F. (1997) Quels schemas numeriques pour les volumes finis INeuvieme seminaire sur les fluides compressibles, CEA Saclay, France.

[EGH 99]

EYMARD, R., GALLOUET, T. and HEREIN, R. (1999), Convergence of finite volume schemes for semilinear convection diffusion equations, Num. Math., vol. 82, p. 91-116.

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[EGH]

EYMARD, R., GALLOUET, T. and HEREIN, R., Finite volume approximation of elliptic problems and convergence of an approximate gradient, submitted to publication.

[ROT 91]

ROBERTS, J.E. and THOMAS, J.M. (1991), Mixed and hybrids methods, in Handbook of Numerical Analysis II (NorthHolland, Amsterdam) 523-640.

[VAS 98]

VANSELOW, R., SCHEFFLER, H.P. (1998), Convergence Analysis of a Finite Volume Method via a New Nonconforming Finite Element Method, Numer. Methods Partial Differential Eq.,U, 213-231.

Finite volume approximation of elliptic problems with irregular data Thierry Gallouet and Raphaele Herbin Umversite d'Aix-Marseille 1, Centre de mathematiques et mformatique 39 rue Joliot Curie, 13453 Marseille, France [email protected], [email protected]

abstract We prove here the convergence of a cell-centered finite volume scheme for the discretization on a non-structured grid of the Laplace equation with irregular data towards the weak solution of the equation. Keys words irregular data.

Finite volumes scheme, non-structured mesh, diffusion

equation,

1. Introduction We are interested here in proving the convergence of the finite volume method in the case of the following model equation:

with Dirichlet boundary condition:

where Assumption 1 /. Q is an open bounded polygonal subset o/IR d , d = 1 or 3, 2. n £ LP(Q) for p 6 [1, +00] or p is a signed bounded measure. Such problems arise for instance when modelling heat transfers in the presence of electric current in which case the heat term due to ohmic loss writes

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where a £ L°°(£7) is the electric conductivity and $ £ Hl(Q) is the electric potential; hence fj, £ L1^) (see e.g. [FH 94]). Another field where such a problem arises is in oil reservoir simulation, where the dimension of the well is often small enough with respect to the size or the domain of simulation so that it is modelled by a Dirac measure in the two-dimensional case (d = 2). The purpose of the proposed presentation is to show that the finite volume method is well adapted to this type of problem; we can show in particular that the analysis tools recently developped by Boccardo, Gallouet et a/. [BG 89] for the study of nonlinear partial differential equations with measure data can be adapted to show the strong convergence as the size of the mesh tends to 0 of the approximate finite volume solution in WQ 'p for any p £ [ towards a weak solution of (l)-(2) which is a function u from Q to IR satisfying:

Remark 1 The Laplace operator is considered here for the sake of simplicity, but more general elliptic operators are possible to handle, for instance operators of the form —div(a(u)Vu] with adequate assumptions on a. A by product of the convergence analysis which is presented here is the existence of a solution of (3).

2. The finite volume scheme The finite volume scheme is found by integrating equation (1) on a given control volume of a discretization mesh and finding an approximation of the fluxes on the control volume boundary in terms of the discrete unknowns. Let us first give the assumptions which are needed on the mesh. Definition 1 (Admissible meshes) Let Q be an open bounded polygonal subset o/IR d . An admissible finite volume mesh ofQ, denoted by T, is given by a family of "control volumes", which are polygonal convex subsets of £l (with positive measure), a family of subsets of Q, contained in hyperplanes of IR , denoted by £ (these are the edges of the control volumes), with strictly positive (d— l)-dimensional measure, and a family of points o/Q denoted by P satisfying the following properties (in fact, we shall denote, somewhat incorrectly, by T the family of control volumes): (i) the set of all control volumes is a partition oftl;

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(ii) For any K G T, there exists a subset £K of £ such that dK = K \ K — (Hi) For any (K, L) G T2 with K ^ L, either the (d-l)-dimensional Lebesgue measure of K fl L is 0 or K D L = W for some cr G £, which will then be denoted by K\L. (iv) The family P = (XK)K^T is sucn that XK G K (for all K G T) and, if a — K\L, it is assumed that XK ^ XL, and that the straight line T>K,L going through XK and XL is orthogonal to K\L. In the sequel, the following notations are used. The mesh size is defined by: size(T) — sup{diam(/\), K G T}. For any K G T and cr G £, m(/\) is the d-dimensional measure of K and m(cr) the (d — l)-dimensional measure of a. The set of interior (resp. boundary) edges is denoted by £-m^ (resp. £€xt), that is £int = {cr G £', cr <£ <9Q} (resp. £ext = {cr G £] cr C d£l}). The set of neighbours of K is denoted by M(K), that is tf(K) = {L G T; 3

We may now introduce the space of piecewise constant functions associated with an admissible mesh and some "discrete W0'p" norm for this space. This discrete norm will be used to obtain some estimates on the approximate solution given by a finite volume scheme. Definition 2 (Discrete norm) Let £7 be an open bounded polygonal subset o/IR , d — 1 or 3, and let T be an admissible mesh. Define X(T] as the set of functions from fi to IR which are constant over each control volume of the mesh. For u G X(T), and p G [l,+oo), define the discrete WQ'P norm by

where, for any cr G T,

where UK denotes the value taken by u on the control volume K and the sets £, £i n t, £ext and £K are defined in Definition 1.

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Let T be an admissible mesh. Let us now define a finite volume scheme to discretize (l)-(2). Let (UK}K£T denote the discrete unknowns associated with the control volumes K £ T- In order to describe the scheme in the most general way, one introduces some auxiliary unknowns namely the fluxes FK^, for all K £ T and

where FK,O is defined by

and Note that the values ua for a £ £;nt are auxiliary values which may be eliminated so that (5)-(8) leads to a linear system of N equations with N unknowns, namely the (UK)KZT, with TV = card(T).

3. Existence and estimates for the approximate solution Let us first prove the existence of the approximate solution and an estimate on this solution. This estimate will yield convergence thanks to a compactness theorem which we recall below. Lemma 1 (Existence and estimate) Under Assumptions 1, letT be an admissible mesh in the sense of Definition 1, and let:

then there exists a solution (UK)K£T t° the system of equations (5)-(8). Furthermore, let p £ [1, ^rj), and let uj- £ X(T) be defined by u-f-(x] = UK for a.e. x £ K, and for any K £ T; there exists C £ IR, only depending on Q, C, p and fj,, such that

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PROOF of Lemma 1 The existence and uniqueness to the solution of the scheme was proved in e.g. [H 95]. Let us now turn to the estimate. For 0 £ (l,+oo), let

0 such that

By a discrete integration by part and from the fact that

one obtains:

where

if

and

if Now for

by Holder's inequality and from (11)

Let us reorder the summation over the control volumes in the right-hand-side and remark that by definition of C, one has da < -J^-. This yields the existence of d € IR+ depending only on yu, D, 0,p and (", such that

Using a discrete Sobolev inequality (the proof of which is similar to the one proved in [EGH 97] or [CGH 98]), there exists Ci £ IR+ depending only on p such that:

From (12) and (13), there exists C% £ IR+ depending only on d,p and C such that

Hence, for one h a s s o that one may choose 0 £ (1,2) such that 2^- < p*. Since p > |p, from (14) and (12), there exists C depending only on //,£}, p and (", such that:

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4. Convergence Let us now show the convergence of approximate solutions obtained by the above finite volume scheme when the size of the mesh tends to 0. One uses Lemma 1 together with the Kolmogorov compactness theorem given at the end of this chapter to prove the convergence result. In order to use the Kolmogorov compactness theorem, one needs the following lemma. Lemma 2 (Estimate on the space translates) Let fi be an open bounded set of JR d , d = 2 or 3. Let T be an admissible mesh and u G X(T). One defines u by u = u a.e. on Q, and u = 0 a.e. on IR \£l. Then there exists C > 0, only depending on Q, such that

The proof of this lemma is an easy adaptation of the proof which is available in [EGH 97] of [EGH 99]. We are now able to state the convergence theorem. Theorem 1 (Convergence) Under Assumption 1, let T be an admissible mesh. Let (UK}K^T be the solution of the system given by equations (5)-(8). Define u-j- G X(T] by u-j-(x) = UK for a.e. x G K, and for any K G T. Let (T^)neiN be a sequence of admissible meshes such thatsize(Tn} —> 0 as n —>• +00 and such that

then there exists a subsequence of (^T n )n€iN; still denoted (w7- n ) n g^, which converges in LP(Q) for p < -£^ to a weak solution u G H1< q<_d_WQ'q(£l} of Problem (3) as size(T) -»• 0. Remark 2 In the case of the uniqueness of a solution to (3), for instance if d = 1, or if d — 3 and f2 is convex (see e.g. [G 97]), then the whole sequence ( w T n )neiN tends to the solution of (3), and therefore u-j- —>• u in Lp(£2) as size(T) —>• 0 under the condition that there exists zeta > 0 such that £7- = mincer m'ma€£K ^- > C for all T. PROOF of Theorem 1 Let Y be the set of approximate solutions, that is the set of functions u-jas defined in Theorem 1 where T is an admissible mesh which satisfies (16). Thanks to Lemma 1, for any p < j^, there exists C\ G IR, only depending on £},//, Co and p, such that ||ur||Lp(n) < C\ for all 1*7- G Y. Then, thanks

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to Lemma 2 and to the Kolmogorov compactness result (see e.g. [EGH 99] or [EGH 97] for the case p = 2), the set Y is relatively compact in L p (0). Now by Lemma 1, we know that for any q E [1, jry), there exists 62 G K, only depending on Q, /LX, ("Q and q, such that ||UT||I, 0) belongs to W 0 ' 9 (fi). Therefore, there remains to prove that if (wT n )neiN C Y converges towards some u G WQ'q(Q) in Lp(Cl) and size(T^) —> 0 (as n —> oo), then w is a solution to (3). We prove this result below, omiting the index n, that is assuming uj- —> « in L p (il) as size(T) -* 0. Let -0 G C^°(ri) and let size(T) be small enough so that 4>(x] = 0 if x G A' and K G T is such that 5A' DdQ ^ 0. Multiplying (5) by ^(XK], and summing the result over A' G T yields

Since the set {A', K G T} is a partition of fi,

where ^7- is defined from Q to H by ~4>T(X} — ^(XK) for x G A'; since ^ G C,?0^), by the Lebesgue dominated convergence theorem, one has:

Now, using the same technique as in the variational framework (see [EGH 97] or [EGH 99]), one has:

Hence, letting size fies

satis-

which, in turn, yields (3) thanks to the fact that u E W0'P(Q), and to the density of Cc°°(n) in W 0 1>9 (n). This proves that uj- —)• u in Lp(£i) as size(T) —>• 0, where u is a solution (in to (3) and concludes the proof of Theorem 1.

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References [BG 89]

BOCCARDO L. andT. GALLOUET, Nonlinear elliptic and parabolic equations involving measure data. J. Fund. Anal., 87, 1, , 149-169, 1989.

[BGV 93] BOCCARDO L., T. GALLOUET, J.L. VAZQUEZ, nonlinear elliptic equations in Rn without growth restrictions on the data, Journal of Differential Equations, Vol. 105, 2, 334-363, 1993. [CGH 98] COUDIERE Y., T. GALLOUET and R. HEREIN , Discrete Sobolev inequalities and Lp error estimates for approximate finite volume solutions of convection diffusion equations, Prepublication 98-13 du LATP, Marseille, submitted. [G 97]

GALLOUET T, Problemes elliptiques et paraboliques non lineaires, Cours de DEA de I'ENS Lyon, 1997.

[EGH 97] EYMARD R., T. GALLOUET and R. HERBIN , The finite volume method, Prepublication 97-19 du LATP, Marseille, to appear in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds. [EGH 99] EYMARD R., T. GALLOUET and R. HERBIN , Convergence of a finite volume scheme for semilinear convection diffusion equations, Numer. Math., 82, 91-116, 1999. [FH 94]

FIARD, J.M., R. HERBIN (1994), Comparison between finite volume finite element methods for the numerical simulation of an elliptic problem arising in electrochemical engineering, Comput. Meth. Appl. Mech. Engin., 115, 315-338.

[GHV 99] T. GALLOUET and R. HERBIN M.H. Vignal, Error estimates for the approximate finite volume solution of convection diffusion equations with Dirichlet, Neumann or Fourier boundary conditions, Prepublication 99-05 du LATP, Marseille, submitted. [H 95]

HERBIN R. (1995), An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh, Num. Meth. P.D.E. 11, 165-173.

Analysis of a finite volume scheme for reactive fluid flow problems

Astrid Holstad Institute of Energy Technology P.O.Box 40 N-2007 Kjeller, Norway Ivar Lie Norwegian Meteorological Institute P.O.Box 43 Blindern N-0313 Oslo, Norway

ABSTRACT We analyze a mathematical and numerical model for reactive multiphase fluid flow problems focusing on chemical reactions and transport of solutes. The discretization consists of a cell-centered finite volume scheme in space and an implicit Runge-Kutta scheme in time. Convergence results are presented for the system of governing equations consisting of both timedependent PDEs and ODEs. Key Words: PDEs, Reactive fluid flow, Time integration

1. Introduction

Most of the geochemical and biogeochemical phenomena of interest to Earth scientists are the result of the coupling of some combination of fluid, heat and solute transport with chemical reactions. Examples of applications where these phenomena are studied are pollution of the atmosphere and the ocean, groundwater contaminant transport, remediation processes, geochemical weathering, ore deposition and diagenesis of sedimentary rocks. In the above applications the chemical reactions and the fluid flow may take place in a very heterogeneous medium under varying temperature and pressure conditions with different time scales. Many different types of chemical reactions take place in a geochemical system:

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Precipitation and dissolution of solids, dissolution and exsolution of gases in to the aqueous phase, ion exchange, redox reactions, decomposition of organic material, radioactive decay, isotope fractionation and biological processes. Because of the complexity of reactive transport processes in porous media, modeling their dynamic behaviour is a challenging task. Many models of different geochemical processes have been proposed over the years, see [HOL 99a] for a brief review. In this paper we analyze a mathematical and numerical model for reactive fluid flow in porous media focusing on efficient integration of the equations over long time periods. As an example, studying diagenetic processes in sedimentary rocks often requires simulation periods of 10 - 100 million years. The model is discussed in detail in [HOL 99a] and [HOL 99b]. The rest of the paper is organized as follows: In section 2 we present a simplified system of the governing equations which discretization is discussed in section 3. The error analysis of this discretization is then discussed in section 4. 2. The governing equations The transport of solutes in a porous medium under the assumption of multiphase flow is given by

where fi is the molal concentration of the ith species, pi is a function to correct for porosity 0, phase saturation Sn and phase density p^, Jj denotes the flux, Ir denotes the rate of the rth chemical reaction, Vir denotes the stoichiometric coefficient and qi denotes any general source term. Let c = [PI/I, • • • ,PN/N]For kinetically governed reactions,

The complete equation system describing multiple species, multiphase and nonisothermal reactive fluid flow becomes when classifying the species into primary (p), secondary (s) and kinetic (k) species,

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where V and VV denote matrices derived from the stoichiometric coefficients in (1), [cp cs Cfc] = [pi/i, • . . , P N / A T ] > Keq,r denotes the mass action equation for the rth equilibrium reaction, as,r and Oj denote species activities, i.e. the concentrations corrected for nonideal behaviour and VIT denotes an element of the matrix V. By splitting the flux into a convective and a diffusive part we can write (3a) in component form,

where £m = 1 for mobile species and 0 for immobile species. Immobile primary or secondary species may occur in applications involving sorption and ion exchange reactions. The convective and the diffusive flux terms in (4) cannot be simplified due to the difference in the phase specific parameters. In applications involving only a single moving phase and no immobile species (4) simplifies to

when defining

3. The discrete system In this section we discretize the simplified equation (5) in space using FVM and in time using the Runge-Kutta formulas

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By augmenting the Butcher matrix A = {ars} with the vector {bs} as the bottom row, the formula (7b) is included in (7a) if we take r = 1 , . . . , P + I and CP+I = 1. In the following we will use only (7a) with this convention. A comprehensive treatment of RK methods can be found in [HWN 87] and [HW2 96]. Discretization of (4) in the general case is treated in [HOL 99a]. Integration of (5) in terms of a finite volume V gives the following ODE,

with convection term

and diffusion term

We integrate the terms in (8) over a time interval h — tn+i — tn and obtain the following expressions: Accumulation,

convection,

diffusion,

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source term, using

We combine corresponding terms and obtain (P + 1) • Np generalized concentration equations and (P + 1) • Rkin mineral equations:

The concentrations of the primary and the secondary species are found from the algebraic system:

For details in solution of the equation system (17a), (17b), see [HOL 98]. 4. Error analysis

In this section we give some error estimates for the finite volume RK discretization presented in previous section. For the transport equations the analysis will be based on the results in [EGH 99], in which references to related

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analysis based on different techniques, e.g. the finite element techniques, also can be found. For the mineral ODEs the analysis will use the theory of numerical methods for ODEs. The transport equations are given in (5) with the definition (6). The dissolution and precipitation of the minerals is assumed governed by first order kinetics as in (2):

when setting y^ = psfNf+i- The relation between \I>j, j = 1,...,JV P and cp is given by the equations (17a) and (17b). For our purpose, it is sufficient to assume the existence of a sufficiently smooth (at least continuous) mapping $ such that cp — 4 > ( \ I > i , . . . , ^N ). We also impose a boundedness restriction on (18), which means that the right hand side of (18) is Lipschitz continuous. The error analysis will consist of deriving a bound on the sum of the discretization errors of \Pj and yi since the equations involved are coupled. We will use the following notation for tyj-. The space and time continuous quantity appearing in (5) is denoted by ^ ! j ( x , t ) . We can project this function onto the space of piecewise constant functions on the cells, Po(T), where T denotes the set of cells covering ft, and this projection on a cell V is denoted by ^jv(t). The fully discrete value on a cell V at timepoint tn is denoted by \IJjy. The total error on V at tn is then but it may also be useful to consider the discretization errors in space and time separately, In the above error expressions xy denotes the representative point on cell V, e.g. the barycenter. The immobile species are computed only on the cells, so the discretization error on a cell V at timepoint tn is For the error analysis of the numerical solution of ODEs by Runge-Kutta methods we use the B-convergence theory, see e.g. [HW2 96] for details. 4.1 Estimates using the theory of Gallouet et al We will use the analyses in [EGH 99], and in order to do that we partly decouple the equation system (5), (18). We insert the relation /; = ^((l-0)yi) into (5) and move the resulting term to the left hand side. We can then write,

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where L denotes the flux operator, i.e. This is a 1-1 mapping between o-j and $j, j = 1 , . . . , Np provided 0 ^ 0 . Thus we have

since L(yi) = 0. The decoupled PDE system (20) is of the type considered in [EGH 99, ch.4]. We equip (20) with an initial condition o~j(x,Q) = O~JQ(X) and a nonhomogeneous Dirichlet boundary condition <jj = Gj on the boundary dft. Consider first the backward Euler finite volume discretization of (20),

where

and of

is the linear operator representing the finite volume discretization

by the upstream scheme for the first term and the VF9 scheme for the second term. We then have the following convergence theorem, Theorem 1 Let 17 be a polygonal open subset of IRd; T > 0, o~j € C2($l x M, H); Ujo € <7 2 (fJ,IR), cjj € C 2 (fj x H, IR), and let T be an admissible mesh in the sense of Definition 3.1 in [EGH 99]. Then there exists a unique vector (ajv)veT satisfying (21) such that

Furthermore, let

where k = size(T], the size of the largest cell in T. Proof: See [EGH 99, theorem 4.1, pp.97-99].« We will now extend this result to higher order time integration methods. It is to be emphasized that this is done primarily to be able to solve the stiff system (18) with a more robust method, including local error control and variable timesteps. The higher accuracy in time is not considered important to reduce

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the the total discretization error. The space error will be of first order because of the projection onto the space of constant functions on the cells, and this error will probably dominate in the total error except for very fine meshes. There is a fairly straightforward extension of theorem 1 to the use of the Backward Differentiation Formulas (BDF) class of methods, see [HWN 87, ch. III.l] for details. The result is: Corollary I Let a, 6 C^p+^(^l x K,1R) and ajo € C(p+l)(Cl,TR). Otherwise the assumptions are as in theorem 1. If we apply a BDF method of order p to (20), the error satisfies

Proof: See [HOL 99a, Sect.5]. • We now proceed to the implicit RK methods. Corollary 2 Let aj <E C( p+1 )(ft x H,IR), aj0 G C(P +1 )(ft,IR), and make the remaining assumptions as in theorem 1. If we apply a P-stage implicit RK method of order p to (20), the error satisfies

Proof: See [HOL 99a, Sect.5]. • Because of the simple relation between \I>j and o~j it follows that the above results, theorem 1, corollary 1 and corollary 2 also apply to tyj. Remark 1: If the exact solution is less smooth than stated in the above results, the convergence order will be correspondingly reduced. If GJ G C^ r+1 ^, r < p the convergence order will be r. 4.2 Semigroup estimates We may also prove convergence results with a more abstract approach using the fully discrete approximation of semigroups acting on a Banach space X. The classical semigroup approach can be used if the generator of the semigroup (the spatial part of the differential operator) is time-independent. The extension to time-dependent differential operators is discussed in [FAT 83, ch.7]. If we start with the Cauchy problem for the transport equations,

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we can construct approximations to the semigroup generated by L. The operator generates a semigroup provided it satisfies the conditions in the Hille-Yosida theorem, see e.g. [FAT 83]. In our case it can be shown that the operator L generates a (7°-semigroup, i.e. a semigroup S(t) which satisfies the basic estimate \ \ S ( t ) \ \ L ( X ) < Mewi, t > 0, see [EVA 98, p.421] . Hence the solution is We introduce some notation for approximation of the semigroup S ( t ] , referring to [MBI 98]. Let XN denote the space of discrete values, in our case this is the space PQ(T). Note that in view of theorem 1, X may be taken as C2. Now define restriction and prolongation operators,

and from the finite volume approximation we have

Let r(z) denote a rational complex approximation to ez, such that r(Lt) approximates the semigroup. The approximation is said to be A-acceptable if r(z] is analytic for 9?(z) < 0 and |r(iy)| < 1, y G IR. Let e be the maximal value such that r(z) is analytic in the set {z : 3ft(z) < £}. We then have the following convergence theorem, Theorem 2 Let r(z) be an A-acceptable approximation to ez of order p, and assume that 0^0 £ £)(!/). Then for

and

provided 0 Proof: The first term representing the error in time is proved in [MBI 98]. A complicated second term representing the error in space is also derived in [MBI 98]. In our case this error can be written

The operators pw and TN are assumed to be uniformly bounded. The expression ||r(/i^)n|| is also bounded since \\r(hA}n\\c(X) < ew^nh\ see [MBI 98]. From (29) the last term in the above space error is O(k). • Remark 2: Initial boundary value problems can be solved by introducing the boundary condition in the definition of D(L), for example we may take

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The non-homogeneous case can be treated with the same tools as we applied in theorem 2. Consider again (28) . The solution is

where * is the convolution operator in time. Denote by &j the solution of the homogeneous problem, then CTj(t} &j(t} -— ffj(t] ffj(t} = = S S * * (Jj(t) (jj(t} if if &JQ CTjQ = = VJQ. CTjO- Let the discrete solutions be given by aa^ j ( t } ) . Hence by !vv — ~ a^ °jVv = p^r^(S PNrN(S * qqj(t)}. subtraction,

The last term on the right hand side can be bounded by O(k] in the X-norm, and the error in the non-homogeneous problem is also given by theorem 2.

5. References [CRM 84]

CROUZIEX M, MIGNOT A., Analyse numerique des equations differentielles, Masson, Paris, 1984.

[EGH 99]

EYMARD R, GALLOUET T, HERBIN R., Finite Volume Methods, In Handbook of Numerical Analysis, Vol VIII, P.G.Ciarlet, J.-L. Lions eds., North-Holland, Amsterdam, 1999.

[EVA 98]

EVANS, L C., Partial Differential Equations, American Mathematical Society, Providence, 1998.

[FAT 83]

FATTORINI H O . , The Cauchy Problem, Addison- Wesley, London, 1983.

[HOL 98]

HOLSTAD A., Numerical solution of nonlinear equations in chemical speciation calculations, To appear in Computational Geosciences, 1998.

[HOL 99a]

HOLSTAD A., A mathematical and numerical model for reactive fluid flow systems, Submitted to Computational Geosciences, 1999.

[HWN 87]

HAIRER E, WANNER G, NORSETT, S P., Solving Ordinary Differential Equations I, Nonstiff Problems, Springer Verlag, Berlin ,1987.

[HW2 96]

HAIRER E, WANNER G., Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer Verlag, Berlin, 1996.

[MBI 98]

BJ0RHUS M, LIE I., Fully discrete rational approximation to semigroups, preprint, DNMI, 1998.

[HOL 99b]

HOLSTAD A., Numerical solution of the reactive fluid flow equations, Preprint, IFE, Kjeller, Norway, 1999.

Convergence of a finite volume scheme for a nonlinear convection-diffusion problem Anthony MICHEL, C.M.I. Marseille. Email : [email protected] ABSTRACT We construct a time explicit scheme for a nonlinear convection diffusion problem which is L°° stable under a C.F.L. condition. We also obtain discrete estimates that allow us to apply Kolmogorov's theorem. Then up to a subsequence we get the convergence of the scheme to a weak solution of our problem. In the last part, we give some elements of a simple proof of uniqueness in a special case by a dual method. Key Words: finite volumes, parabolic nonlinear equations , Kolmogorov's theorem. 1. Problem

We consider the following nonlinear parabolic degenerate problem in a bounded polygonal domain

This problem arises in the study of a two-phase flow in a porous medium. The unknown u is the saturation of the first phase so it takes its values in the interval [0, 1]. The global flux vector v is a given function in C1^ x [0,T]). In the same way, / and (p are given in ^([O, !])• The only assumptions that we make are

• (H3) tp is strictly increasing but

Let T a cell-centered unstructured mesh (see for instance [EGH 97], [EGH 99], or [YNS 97] for definition of an admissible mesh) and St a time step. We will always suppose for the sake of simplicity that there exist N £ N such that (N + l)St = T. We construct an approximate piecewise constant solution a.e. on Q x [0,T) by

where {u^} is defined by an "upwind" finite volume scheme which we now describe.

174

Finite volumes for complex applications For a control volume K E T we take

For K G T and n <E [0, N\ we define w^+1 from {unK}K^r by

Vn

r

i

r_j\

-f

r> ,-> r

n~<

m(KnL)

i

n ffi •

•

j

K,L = Ja v ('' n(") ' n^,£ if

If we denotes by x~ = max(0, —x), equation [3] is therefore equivalent to the following non-conservative form

By using monotony of the scheme we then easily prove the following lemma Lemma 1

Assume that

Then the scheme is L°° stable, i.e. if UQ £ [/4,-B] a.e. then UT,st £ [A, B] a.e..

3. Discrete estimates on

then there exist (7(£,«o) > 0 such that

and

Proof. We use again the non conservative form [4]. We multiply the equation by 8tunK and sum over K 6 T and n G [0, N}. We get EI + E2 + E3 = 0, where

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Let give now a usefull result (see for instance [EGH 97]) Lemma 2 (technical lemma) Let g(x) tone then if we denote by

where for all a and b

By lemma 2,

But

and finally

So by using proposition 1, we get

is mono-

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Finite volumes for complex applications

It remains to estimate R\

Therefore by using condition [6], we get

Moreover,

Then by collecting the previous inequal-

ities, we complete the proof with 4. Translates estimates

As a consequence of estimate [7], by classical technics that can be seen for instance in [EGH 97], we get the two following inequalities Corollary 1 (Space-translate estimate) Under the assumptions of proposition 2,

where Corollary 2 (Time-translate estimate) Under the assumptions of proposition 2, there exist C'(e,

5. Compacity and Convergence

We handle now the first part of the conclusion under the convergence of the scheme. For that we need to use the classical following compactness theorem.

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Theorem 1 (Kolmogorov) Let F a bounded subset o/L 2 (M r f ) such that

Then for every Q CC K. , 3~ is relatively compact in L 2 (f2). We are now able to state the convergence theorem Theorem 2 (Convergence theorem, part 1) Let (Tm,8tm) a sequence of meshes and time steps that satisfies assumptions of proposition 2. Assume moreover that hm tends to zero when m tends to oo (which implies that 6tm tends to zero ) . Then there exist u E £°°(ftx(0,T)) such that tp(u) € L2(0,T, Hl(ty) and up to a subsequence, limm-^oo v>Tm,6tm — u for L°°(Q x (0,T)) weak star topology and in LP(fi x (0,r)), Vp< oo. ' Elements of proof. Let extend uj-^t on M9*1 by zero out of f2 x (0,T). From corollary 1 and corollary 2, we directly deduce that for every (£, s] E M 9+1 ,

where This inequality allow us to apply theorem 1 and we obtain regularity on the limit by looking at accroissements taux which converge to the derivatives in T>'. (see for example [EGH 97]) 6. Convergence, part 2 Theorem 3 (Convergence theorem, part 2) We suppose that the assumptions of theorem 2 are satisfied, and we assume also that there exists 9 > 0 such that for all mesh Tm, the following regularity property is satisfied :

Then the function u given in theorem 2 is solution of [1] in the following sense : W E Ctest = {n e ,T) = 0},

Proof. The convergence of un to u is strong which implies that f ( u n ] and (p(un} converge to f ( u ) and (p(u). So it suffices to show that

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Finite volumes for complex applications

Let then multiply Let ra fixed and T = We get and sum over K £ T and n where

where

and

where ^nKQL is equal to ^nK if v^ given by

L

< 0 and ^nL otherwise. We compare Tz- to 5,-

and

Classically, (see for instance [YNS 97], [EGH 97]) because ip is a regular function and V(T) = 0 we get lim (Si - 7\) = 0. m—>oo

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By using estimates [8], [7] and regularity condition [9] we get also

and Then

and the proof is complete.

7. Uniqueness

Let

where

be two solutions of

We denote by Ud = wi — ^2- For all

and

So it is natural to pay

attention with the dual problem :

From [LSU 68], we can state the following result Theorem 4 (Existence to the regularized dual problem) Let F , v and <£ be C°° functions under Cl x [0,T], and assume that there exists 6 > 0 such that 3>(x,t) > S. Then for every x £ C^ (Q x (0,T)) there exists an unique solution to [12] Moreover, we have also the following estimates Proposition 3 Let -0 a solution to the regularized dual problem with second member xana Mx, M$ , Mv and Mp some upper bounds for \x\, $, |v| et \F\. Then there exist C(x, M<j>, Mv, MF, &, T) > 0 such that

and

Elements of proof. [13] is a direct consequence of the maximum principle for parabolic equations. For [14] and [15], we multiply the equation by A?/> and integrate over Q x (0, T). Because of [13], |V^||L2(n x (o,T)) is controled by >/||AV>||L3(nx(o,T))- We complete then the proof by using time and space integrates by part and Young inequalities. We are now able to give the main result of this section

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Finite volumes for complex applications

Theorem 5 Assume if 1 is an holder continuous function with exponent ^. Then there exist an unique solution to [10]. Proof. By theorem 3, there exist at least one solution to [10]. We now turn to the study of uniqueness. Let x £ C™(ft x (0,T)) and S > 0 (6 < M$). <J><5 = max(S, <2>) is again in L°°(fi x (0,T)) and S is a lower bound for it. We don't have regularity hypothesis on <$<j and F but we can construct Gn, Fn and \n some sequences of regular functions on fi x [0,T] that converge to $5, F and v in Lp(£l x (0,T)) for p < oo and such that

For every n, by theorem 4 there exist a solution ipn in (7 2>1 (fi x [0,T]) to the dual problem associated to Gn, vn, Fn and x- Because the upper bounds of Gn, vn, Fn and the lower bound 8 of Gn are independant from n, estimates on Ai/>n and VV'n are also independant from n, so we get

and But denote by

Because Moreover

because

and are equal on we get

is an holder continuous function with exponent tends to zero, so that

Then, if we

ud

on

Since that is true for every regular function x, the proof is complete. [EGH 97]

R. EYMARD, T. GALLOUET, R. HEREIN Finite volume methods, Prebublication 97-19 LATP Marseille, to appear in Handbook of numerical analysis, Ph. Ciarlet &; J.L. Lions ed., 1997.

[EGH 99]

R. EYMARD, T. GALLOUET, R. HEREIN Convergence of finite volume schemes for semilinear convection diffusion equations. Numerische Mathematik 82 : 91-116, Springer Verlag, 1999.

[YNS 97]

Y. NAIT SLIMANE Methodes de volumes finis pour des problemes de diffusion-convection non lineaires. These de 1'universite Paris 13, 1997.

[LSU 68]

O.A. LADYZENSKAJA, V.A. SOLONNIKOV, N.N. URAL'CEVA. Linear and quasi-linear equations of parabolic type. Transl. of math, monographs 23, American Mathematical Society, 1968.

Convergence analysis of a cell-centered FVM

Hans-Peter Scheffler*, Reiner Vanselow** *Institut fur Analysis **Institut fur Numerische Mathematik TU Dresden, Mommsenstr. 13 D - 01062 Dresden

ABSTRACT A well-known cell-centered FVM with Voronoi boxes for discretizing the Poisson equation is analyzed. To achieve this purpose, a nonconforming FEM is constructed, such that the system of linear equations obtained by using the nodal basis coincides completely with that for the FVM. In this way, convergence properties of the FEM, which are formulated in terms of function space norms, can be transformed to the FVM. Key Words: cell-centered finite volume method, Voronoi boxes, convergence analysis, nonconforming finite element method.

1. Introduction Finite Volume Methods (FVMs) are standard methods for finding numerical solutions of partial differential equations. Like Finite Element Methods (FEMs) they can be applied to a wide class of problems over arbitrary domains and allow local refinements of the domain partition. For a given / 6 L^^l] we consider the Poisson equation In order to simplify the presentation we restrict ourselves to open, convex and bounded polygonal domains fi. The convergence proof of the FVM is based on the following two steps: • Description of a nonconforming FEM such that the system of linear equations coincides completely with that for the FVM. • Proof of convergence for the corresponding FEM.

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Finite volumes for complex applications

In distinction to other authors (cf. e.g. [HAC 89]) we use a nonconforming FEM basing on dual Voronoi boxes. The dual Voronoi boxes in combination with the choice of special discrete function spaces are well suited for our aim. The linear convergence of the FEM with respect to some energy norm is proved under the assumption that the solution u of (1.1) belongs to H 2 ( f t ) . As usual, some geometrical properties have to be satisfied for the partitions of J7. A more detailed representation of this subject is given in [VAS 98]. Here we give an improved version for the estimation of the consistency error term and discuss convection-diffusion problems, too.

2. The FVM and the corresponding nonconforming FEM

2.1. Box and dual box partitions In the following, let M = {P} with P 6 cl (fi) be an arbitrary finite set of points. Further, we use the notations Mi = M n fi and Mb — M n F, where m = card (Mi) > 0 and card (Mb) > I have to be satisfied. Let \P — Q\ denote the Euclidian distance between two points P and Q. Definition 1. For P £ M the Voronoi box bp is defined by The set B = {bp} of all Voronoi boxes is called box partition. If for different points P, Q £ M the intersection bp D bq is non-empty, then the corner points are denoted by Ei(P,Q) and E%(P, Q), i.e. it holds E1(P,Q)E2(P,Q) = bPnbQ. Now, for P £ Mb we define the set NE(P)

= {Q £ Mb : P and Q are neighbours on F},

and for P £ Mi we use the notations

To define the FEM, which is used for the convergence analysis of the FVM, we need another partition of the domain fi which is dual to the box one. Definition 2. For P £ Mi and Q £ NN(P) defined by

the dual Voronoi box dbpQ is

The set dB = {dbpq} of all dual Voronoi boxes is called dual box partition.

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For the further considerations, the following property is assumed to be satisfied: For all P G Mb and Q e NE(P} it holds which is obviously equivalent to

2.2. The Finite Volume Method If we integrate both sides of (1.1) over the Voronoi box bp £ B we obtain by applying Green's formula the equations

where the vector n(bp) denotes the outer normal direction of bp . Further, on the straight lines bpCibQ the outer normal n(bp) coincides with the vectors

Now, if in (2.1) the arising integrand [epg]Tgrad u is substituted by the constant finite difference approximation

and if (A') is used, then we obtain the following well-known cell-centered FVM for the Poisson equation (1.1): Find uv = uv(M) G Rm such that where the matrix Lv and the right-hand side bv are given by

and

otherwise The point P belongs to the index k and the point Q to the index /. 2.3. The corresponding Finite Element Method A weak formulation of the boundary value problem (1.1) reads as follows: Find u such that

184

Finite volumes for complex applications For the FEM we define a finite-dimensional space Vh by

v is continuous in P e M and v(P) = 0 VP e Mb} , where P(P,Q] with P = (xp,yp)T and Q = (xQ,yq)T denotes the space P(P, Q) = span {1, [(zp - Z Q )(Z - XP] + (yP - yQ}(y - yP)]}, and the function values at the points P G Mi are choose as degrees of freedom. For the convergence analysis, we consider the nonconforming FEM: Find Uh = Uh(M] e Vh such that

with

ePQ defined by (2.2) and D(P) defined by (2.1). The bilinear form ah is also defined on [V 0 Vh] x V^ and, because of grad which results in [gradv]Tgr&dwh This implies

Using the nodal basis functions {<£p} and denoting the vector of the function values of the solution Uh of the FEM (2.5) in the points P e Mi with w^, a linear system of equations arises, which has the form The stiffness matrix LE, the vector UE = uE(M] G Rm and the right-hand side bE are given by Here, the indices are analogously used to Section 2.2. From the special form of the right-hand side dh it follows (cf. [VAS 98]) Theorem 1. The problems (2.3) and (2.9) are equivalent, i.e. the vectors uv and UE coincide. 2.4. Convergence concept for the FVM The solution of a FVM is a vector in Rm, whose entries can be considered as approximations of u(P), P € Mi, where u solves (1.1). Nevertheless and

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185

in contrast to some other authors (cf. e.g. [LAM 96]), we prove convergence results for the FVM (2.3) in terms of function space norms. For that reason we need a bijective correspondence between the vector uv G m R that solves (2.3) and the function Uh G Vh which is the solutions of (2.5). Theorem 1 supplies such one additionally satisfying the interpolation property uh(P] = v% for all P £ Mi. Let now a norm H.^ on Vh be given that is a seminorm on V 0 Vh • Definition 3. For a sequence {Mh} of sets of points satisfying the assumptions in Section 2.1 let [uv(M^)} be the corresponding sequence of approximate solutions defined by (2.3). We say that the FVM (2.3) is convergent with respect to \\.\\h , iff \\u — Uh\\h approaches 0 for the solution u of (2.4) and the sequence {uh} = {uh(Mh)} defined by the FEM (2.5). 3. The convergence result

3.1. The first step To prove convergence of the nonconforming FEM (2.5) the well-known second Strang Lemma is used, which leads to an estimation of the form

with a positive constant C independent of h (cf. [VAS 98]). In our application we choose the energy norm with a/! given by (2.6), such that the assumptions of the second Strang Lemma are satisfied. 3.2. The second step To obtain an error estimation it is necessary to deduce bounds for the terms on the right-hand side of (3.1). For the approximation error term we introduce the interpolation operator TldB : H2(ty n ff 1 (fi) C V —>• Vh, which is defined by

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Finite volumes for complex applications

and take advantage of

Standard techniques like that one in [CIA 91] lead to with a positive constant c independent of h, if it holds u 6 H2(ty Pi HQ (fi) and if the partitions of fi satisfy some geometrical properties, which are comparable with the minimal angle condition for a corresponding Delaunay triangulation. The problem, that it holds can be overcome by appropriate spaces and seminorms like and

in place of Sobolev spaces like Hl(dbpQ] as well as a slight modification of Theorem 15.3 of [CIA 91]. Obviously, in that theorem the assumption can be substituted by the weaker one For the consistency error term

we have to estimate which was done in [VAS 98] in the following way: At first, we obtain

with

Thereby, it is used, that the standard bilinear form a^, which is given by the right-hand side in (2.8), is substituted by that one in (2.6). was estimated by applying\8db Theorem 33.1 of [CIA 91], Then PQ which under the same assumptions as above leads to

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again with a positive constant c independent of h. But, if we use (2.7), in place of (3.4) we get with

Now, \rjdpQ(u}\ can be estimated by applying the well-known BrambleHilbert lemma (cf. e.g. Theorem 28.1 of [CIA 91]). Together with

this results in the same inequality (3.5), but gives a shorter proof.

3.3. The third step Altogether (3.1), (3.3) and (3.5) lead to the following convergence result. Theorem 2. // the solution u of (2.4) belongs to H 2 ( o ) D H o ( o ) and if the partitions of o satisfy some geometrical properties (for the details cf. [VAS 98]), then there exists a positive constant C independent of h such that it holds

where \\-\\h is defined by (3.2) and uh is the solution of the FEM (2.5). Because of Definition 3 the convergence properties of the FEM (2.5) can be transformed to the solution of the FVM (2.3).

4. Discussion In [HAC 89] the convergence of a FVM like that, which is given by (2.3), is proved by using triangles for the partition of o and the well-known conforming linear FEM. It is an advantage of our approach, that the analysis can extended to convection-diffusion equations of the form div{— £ grad u + b u} = f, where E is a positive parameter and b is a given constant vector. One possibility is the use of a full-upwind technique with the approximation [ePQ]T {- e grad u + b u}

with the function function K defined by

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Finite volumes for complex applications

V

which altogether leads to a FVM like that in [MIS 98]. Another one is to use exponential fitting with the approximation

(cf. e.g. [BBF 90]), where the Bernoulli function B is defined by

5. Bibliography

[BBF 90]

BANK, R.E., BURGLER, J.F., FICHTNER, W., SMITH, R.K., «Some Upwinding Techniques for Finite Element Approximations of Convection-Diffusion Equations», Numerische Mathematik, 58, 1990, p. 185-202.

[CIA 91]

CIARLET, P.G., «Basic Error Estimates for Elliptic Problems», in Handbook of Numerical Analysis - Vol. II - Finite Element Methods (Part 1), P.G. Ciarlet, and J.L. Lions, Eds., Elsevier (1991), p. 17-351.

[HAC 89]

HACKBUSCH, W., «0n First and Second Order Box Schemes>, Computing, 41, 1989, p. 277-296.

[LAM 96]

LAZAROV, R.D., MISHEV, I.D., in «Finite Volume Methods for Reaction-Diffusion Problems», in Finite Volumes for Complex Applications, (1996), p. 231-240.

[MIS 98]

MISHEV, I.D., «Finite Volume Methods on Voronoi Meshes», Numerical Methods for Partial Differential Equations, 14, 1998, p. 193-212.

[VAS 98]

VANSELOW, R., SCHEFFLER, H-P., «Convergence Analysis of a Finite Volume Method via a New Nonconforming Finite Element Method», Numerical Methods for Partial Differential Equations, 14, 1998, p. 213-231.

Error estimates on the approximate finite volume solution of convection diffusion equations with boundary conditions

Thierry Gallouet, Raphaele Herbin Umversite de Provence, CMI, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France;

Marie Helene Vignal Lab. MIP, UFR MIG, 118 route de Narbonne, 31062 Toulouse, Cedex 4, France; ABSTRACT. We study here the convergence of a finite volume scheme for a diffusionconvection equation on an open bounded set of Rd (d — 2 or 3) for which we consider Dirichlet, Neumann or Fourier boundary conditions. We consider unstructured meshes which include Voronoi' or triangular meshes; we use for the diffusion term a "four-point" finite volume scheme and for the convection term an upstream finite volume scheme. Assuming the exact solution at least in H2 we prove error estimates in a discrete HQ norm of order the size of the mesh. Discrete Poincare inequalities then allow to prove error estimates in the L2 norm. KEY WORDS : boundary conditions, convection, diffusion, error estimate, finite volume schemes.

1. Presentation of the problem Let o be an open bounded subset of R (d = 2 or 3) which is assumed to be polygonal if d — 2 and polyhedral if d = 3. We denote by 9o its boundary and by n the unit normal to 9o outward to 17. We consider the following convection-diffusion-reaction problem:

with different boundary conditions and where Assumption 1 In this paper, we consider three different types of boundary conditions. The first one is a Dirichlet condition. Let gD G H3/2(dQ) (in order to obtain error estimates), then: The second one is a Neumann condition, assuming Assumption 2:

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Assumption 26 = 0 and divv = 0 on Q, v n — 0 on dQ and gN E Hl^(d^l] satisfies the following compatibility relation: fd^9N ( x ) d~f(x) + f^ f ( x ) dx — 0. Finally the last type we consider is a Fourier condition, under Assumption 3:

Assumption 3 gF E Hl/~(dty, A E H such that v.n/2 + A > 0 on dQ. Furthermore, if v(x).n(x)/2 + A = 0 for all x E OQ then one assumes the existence of O C Q such that its d-dimensional measure m(O) ^ 0 and such thatdiv(v)/2 + b^ 0 on O. Remark 1 Assumptions 1 and 3 give the coercitivity of the elliptic operator associated to the vamational equality of (I), (4)- It does not need a compatibility relation, so we consider that, even if X = 0, this case is not a Neumann condition but a Fourier condition. This elliptic problem is then discretized with a finite volume scheme: a "four-point" scheme is used for the diffusion term and an upstream scheme for the convection one. A discrete system is obtained for each type of boundary condition. Existence and uniqueness (for the Neumann's boundary condition, the uniqueness is up to a constant like in the continuous case), of the approximate solution is proven. If the exact solution is assumed to be at least in H 2 (Q), one may then establish the convergence of the scheme by proving error estimates; a first estimate in a discrete HQ norm is obtained. An error estimate in the L2 norm follows with the help of discrete Poincare inequalities. The convergence of the method for Neumann and Fourier conditions requires some additional work compared to that of the Dirichlet case. In the case of Neumann boundary conditions, a "'discrete mean Poincare" inequality needs to be proven in order to obtain an L2 error estimate. In the case of the Fourier condition, it is interesting to note that an artificial upwinding has to be introduced in the treatment of the boundary condition in order for the scheme to be well defined with no additional condition on the mesh. Finite volume schemes for a diffusion convection equation with homogeneous boundary conditions were studied in e.g. [LMS96], [He93] and [VPL92] with different assumptions on the data and the mesh. 2. Discretization

In order to discretize the problem, first we define the mesh. Definition 1 (Admissible meshes) A finite volume mesh of Q, denoted by T, is given by a family of "control volumes", which are open polygonal (or polyhedral) convex subsets of Q (with positive measure), a family of subsets of Q contained in hyperplanes of IR , denoted by E (these are the edges (if d = 1) or sides (if d — ?>) of the control volumes), with strictly positive (d — 1)dimensional measure, and a family of points of f2 denoted by 'P. The finite

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volume mesh is said to be admissible if the properties (i) to (iv) below are satisfied and restricted admissible if the properties (i) to (v) below are satisfied. (i) The closure of the union of all the control volumes is £1; (n) For any K £ T, there exists a subset EK of 8 such that dK = K \ K = (Jae£KW. Let £ = UKET^K(tii) For any (A', L) £ T2 with K ^ L, either the (d— 1}-dimensional Lebesgue measure of K D L is 0 or K Pi L = ~a for some cr £ £, which will then be denoted by K\L. (iv) The family P — (XK)K£T ls such that XK £ K (for all K £ T) and, if a = K\L, it is assumed that XK ^ XL, and that the straight line T>K,L going through XK and XL is orthogonal to K\L. (v) For any a £ £ such that a C dQ, let K be the control volume such that a £ £K- If XK £ &, let T>K,O be the straight line going through XK and orthogonal to a, then the condition T>K,a H cr =£ 0 is assumed; let ya - VKja H (T.

In the sequel, the following notations are used. The mesh size is defined by: size(7~) = supjdiam(A'), A" £ 7~), where diam(A') is the diameter of K £ 7~. For any K £ 7~ and a £ £, m(A') is the d-dimensional Lebesgue measure of K, m(

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where o?7 is the integration symbol for the (d— l)-dimensional Lebesgue measure on the considered hyperplane. Since the approximate solution is constant on each cell of the mesh, one may approximate fK bu(x) dx by frm(A') UK. For all a £ £K, one denotes by FK,O (respectively VK,O] the numerical diffusion (respectively convection) flux. Then the considered finite volume scheme is defined by the following equation:

First, one gives the fluxes for interior edges. Let a £ ZK n£i n t and let us denote by L the cell in T such that a = K\L, one approximates the diffusion flux using a "four-point" finite volumes scheme and for the convective numerical flux, one uses an upstream scheme, that is

where

Now let us give the discretization for boundary terms. Let a £ SK H £ e xt> we consider the three cases corresponding to the different boundary conditions • Dirichlet boundary condition

As u is known on the boundary, we set

and the numerical convective flux VK,<J is defined using (6) with

and ya defined in Definition 1. Remark that the definition of the diffusion flux on a boundary edge (8) allows da — 0, in this case one has UK — gD(ya) and FK^ becomes an unknown. • Neumann boundary condition

Since v • n = 0 on dQ the numerical convection flux equals the exact one, that is 0. We do the same for the numerical diffusion flux, we set

• Fourier boundary condition

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In this case the discretization of boundary terms is performed with the help of some auxiliary unknowns which are defined on the edges of the boundary. These may be eliminated when solving the linear system. We shall however keep them throughout our study because they simplify several expressions in the error estimate. Hence in this section the discrete unknowns are (UK)K£T U (ua)ae£ext- Then ones defines FK a by (8) and VK o by (6) with ua + given by (9). There remains to give the equations associated with the boundary unknowns (ua}a££f^• These are obtained by discretizing (4). The discretization which we choose involves the upstream valued W CT)+ in order for the scheme to be well defined with no additional condition on the mesh. It writes:

Remark 2: In order to discretize the boundary condition on an edge a G £ext of K G 7~, we use a decentered scheme summing and substracting f v-nd^(x). This choice is performed in order to prove existence and uniqueness for A < 0 with no restriction on the mesh; (see Remark 3). In fact, it would be more natural to discretize the boundary condition as follows:

However, if X < 0, the proof of existence, uniqueness and convergence towards the exact solution requires more restrictive assumptions on the mesh. Hence, (12) will be preferred for the discretization of the boundary condition so as to be able to handle negative values of X with no additional condition on the mesh.

3. Existence and uniqueness of the approximate solutions • Dirichlet boundary condition Let us begin with the approximate solution associated to (1) with the Dirichlet boundary condition. The proof of existence and uniqueness of the approximate solution is performed by establishing a discrete maximum principle Proposition 1 Under Assumption 1 and assuming gD G H3/2(dQ), let T be an admissible mesh in the sense of Definition 1. Let (ua)a^£ext, be defined by (10). If fK f ( x ) dx > 0 for all K £ T, and ua > 0, for all *

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where (UK)KZT

ls

the unique solution to (5), (6), (7), (8) and (9).

• Neumann boundary condition

One proves the following result which gives the existence of the approximate solution and its uniqueness up to a constant like in the continuous case. Proposition 2 Under Assumptions 1 and 2, let T be an admissible mesh in the sense of Definition 1. Then, there exists a solution u-j- to (14), (5), (6), (7) and (11). This solution is unique up to a constant. The proof of this result can be found, for instance, in [GHV] or [Vi97]. • Fourier boundary condition

One proves the following result: Proposition 3 Under Assumptions 1 and 3, let T be an admissible mesh in the sense of Definition 1. Then there exists a unique solution (UK}K^T U K) a e £ e x t to (5), (6), (7), (8), (9) and (12). Idea of the proof:

Assuming fK f ( x ) dx for all K £ T and fa (JF (%} d^(x) = 0 for all a £ £ext and using the numerical scheme (for more details see [GHV]) one proves that

where Dau-j- — \UK — UL\ if cr £ ^ n t , a = K\L and Dau-j- = \UK — ua\ if cr £ £ext n £ # , K £ T- Thanks to Assumptions 1 and 3, this result gives UK = 0 for all K £ T and ua = 0 for all a £ £ ext . This proves uniqueness and therefore existence since the dimension of the space is finite. Remark 3: // the discretization (13) is used instead of (12), computations similar to those of the above proof yield:

where VK,O — / v(x) • n(z) d^(x}. So if A > 0, this inequality gives Proposition 3 but if \ < 0 one must assume some more restrictive assumption on the mesh as we already mentionned in Remark 2; for instance one could assume m(cr)A + | fa v(ar) • n(x)d-y(x) + Xdam(a) > 0. We may now define the approximate solution by

4. Error estimates

One gives in this section the three results corresponding to the three different boundary conditions. Theorem 1 (Dirichlet boundary condition) Under Assumption 1 and assuming gD G H3/2(dQ), let T be a restrictive admissible mesh in the sense of Definition 1 and let (," = min^x min CTe £ K 'a^ '. Let u-j be defined by (14), (5), (6), (7). (8) and (9). Assume that the unique vamational solution u to (1) and (2) belongs to H ~ ( $ l ) . Let e-r be defined by e-j-(x) = e^ = U(XK] — UK if x G A', A' G T. Then, there exists C, only depending on u, v, b, Q, d and (,, such that

where Daer = \eK - eL\ if a G £mt, & = K\L, Daer - \CK tf & ^ £ext H SK. Theorem 2 (Neumann boundary condition) Under Assumptions 1 and 2, let T be an admissible mesh in the sense of Definition 1 and let (" = minxeT m in,7££ K ' a T—-. One assumes that the unique variational solution u G Hl(Q), such that f^u(x)dx = 0, of Problem (1), (3) satisfies u G H2(Q). Let UT be the solution to (14), (5), (6), (7) and (11) such that Z^A'eT m(-^') UK — ^L,K£T m (^') U ( X K ] , where XK is defined in Definition 1. Let e-r be defined by e-j-(x) — ex = U(XK) — UK if x G A', A" G T. Then, there exists C, only depending on u, v, b, d, Q and (,, such that: Clfy

where Dae-j- = CK — CL| if & — A"|L and the set £jnt is defined in Definition 1. Theorem 3 (Fourier boundary condition) Under Assumptions 1 and 3, let T be a restrictive admissible mesh in the sense of Definition 1 and let C = mincer min^,. d i a ™j*). Let ur be the solution to (15), (5), (6), (7), (8), (9) and (12). Assume that the unique variational solution u of Problem (1), (4) satisfies u belongs to // 2 (Q). Let e-j- be defined by e-r(x] — eK — U(XK] — UK if x G A, A" G T and eT(x) = ea = u(ya) - ua if x G u, a G £ e xtThen, there exists C, only depending on u, v, b, X, Q and (,", such that:

and where

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Proof of Theorem 3: Using the numerical scheme, the fact that ml. FK,O is a consistant approximation of ^f) fa — Vw(x) • n(.r) d*y(x) for all a € £ and the conservativity of the numerical and exact flux through a given edge, one proves the first estimate of (16), which can be seen as an error estimate in a discrete HQ norm. In ordre to establish the second one, one uses the following discrete Poincare inequalities: Lemma 1 Let T be an admissible mesh in the sense of Definition 1 and u be a function which is constant on each cell ofT and each edge o/£ ext , that is u(x) = UK if x £ K, K G T and u(x) = ua if x (E a, a G £ e xt- Let F C d£l such that its (d — 1}-dimensional measure m(F) ^ 0 and O C Q such that its d-dimensional measure m(O) 7^ 0. Then there exists C, only depending on Q, such that

where for all a G £, D0u is defined in Theorem 3. 5. Bibliography [EGH] R. EYMARD , T. GALLOUET and R. HEREIN , The finite volume method, to appear in Handbook of Numerical Analysis, J.L. Lions and P.G. Ciarlet eds. [EGH99] R. EYMARD, T. GALLOUET and R. HEREIN , Convergence of finite volume schemes for semilinear convection diffusion equations, accepted for publication in Numer. Math. (1999). [GHV] T. GALLOUET, R. HEREIN and M.-H. VIGNAL, Error estimates on the approximate finite volume solution of convection diffusion equations with Dirichlet, Neumann or Fourier boundary conditions, submitted. [He93] HEREIN R., 1993, An error estimate for a finite volume scheme for a diffusion convection problem on a triangular mesh, Num. Meth. in P.D.E.

[LMV96] R.D. LAZAROV, I.D. MISHEV and P.S. VASSILEVSKI (1996), Finite volume methods for convection diffusion problems, SIAM J. Num,er. Anal. 33, 31-55. [VPL92] P.S. VASSILEVSKI, S.I. PETROVA and R.D. LAZAROV (1992), Finite difference schemes on triangular cell-centered grids with local refinement, SIAM J. Sci. Stat. Comput. 13, 6, 1287-1313. [Vi97] M.H. VIGNAL (1997), Schemas Volumes Finis pour des equations elliptiques ou hyperboliques avec conditions aux limites, convergence et estimations d'erreur These de Doctorat, Ecole Normale Superieure de Lyon.

The limited analysis in finite elasticity

Igor A. Brigadnov Department of Informatics North- Western Polytechnical Institute 5, Milhonnaya Str., St. Petersburg 191186, Russia E-mail: [email protected] nwpi.ru

ABSTRACT The variational formulation of the elastostatic bounary-value problem for hyperelastic materials is considered. For elastic potentials having the linear growth in modulus of the distortion tensor, the limited analysis problem is formulated. In the framework of this problem the strength of nonlinear elastic solid is estimated. From the mathematical point of view this problem is non-correct and, therefore, needs a relaxation. The partial relaxation is descibed for the limited analysis problem. It is based on the special discontinuous finite-element approximation with functions having breaks of the sliding type. The numerical results show that this technique has qualitative advantages over standard continuous finite-element approximations. Key Words: elastostatics BVP, the limited analysis problem, partial relaxation, discontinuous FEA.

1. Introduction The solution of elasticity boundary-value problem (BVP) is of particular interest in both theory and practice. At present there are many models of elasticity in the framework of the finite deformations theory (GRE 75, BAR 76, CIA 88, LUR 90). Adequacy and the field of application of every model must be found only by the correlation between experimental data and solutions of appropriate BVPs. Therefore, the analysis of mathematical correctness and the treatment of numerical methods for these problems is very important (BRI 93-98).

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In this paper the finite elasticity BVP is formulated as the variational problem for the displacement (CIA 88). For materials with ideal saturation the elastic potential <&(x, Vit) of which has the linear growth in |Vw|, where x ^ u G R3 is the map, the existence of the limited static load (such external static forces with no solution of BVP) and discontinuous maps with breaks of the sliding type was proved by the author (BRI 93-98). From the physical point of view these effects are treated as the destruction of a solid. The limited analysis problem for nonlinear elastic solid is formulated. In the framework of this problem the estimation from below for the limited static load is calculated. As a result, the shape optimization problem is also formulated for the nonlinear elastic solid of the maximum strength. It is demonstrated, using the simplest example, that the limited analysis problem has discontinuous solutions with breaks of the sliding type. From the mathematical point of view the limited analysis problem is noncorrect (EKE 76, FUC 80, TEM 83, GIU 84) and, therefore, needs a relaxation. We use the partial relaxation which is based on the special discontinuous finiteelement approximation (FEA) (REP 89, BRI 98). After this discontinuous FEA the limited analysis problem is transformed into the non-linear system of algebraic equations which is badly determined, because the global stiffness matrix has lines with significantly different factors. Therefore, for the numerical solution the decomposition method of adaptive block relaxation is used (BRI 96c-98). Its main idea consists of iterative improvement of zones with "proportional" fields by special decomposition of variables, and separate calculation on these variables. The numerical results show that for the estimation of the limited static load, the proposed technique has qualitative advantages over standard continuous finite-element approximations. 2. The limited analysis problem in nonlinear elasticity

Let a rigid body in the undeformed reference configuration occupy a domain Q C R3. In the deformed configuration each point x £ ft moves into a position u(x) — v(x) + x 6 R3, where it and v are the map and displacement, respectively. Here and in what follows we use the Lagrangian coordinates. We consider locally invertible and orientation-preserving maps u : ft —> R3 with gradient (the distortion tensor) Q(u] = VM : ft —» M3 such that det(Q) > 0 in ft (GRE 75, CIA 88, LUR 90), where the symbol M3 denotes the space of real 3 x 3 matrices. The finite deformation of materials is described by the energy pair (Q, S), where £ = {£"} is the first non-symmetrical Piola-Kirchhoff stress tensor. It is known that the Cauchy stress tensor a = {aa^} has the components a01/3 = (det(Q)) -1 Ef Qf. Here and in what follows the Roman lower and Greek upper indeces correspond to the reference and deformed configurations, respectively, and the addition over repeating indeces is used.

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Elastic materials are characterized by the response function (the stressstrain relation) £ = E(x, Q) such that E(a?, I) = O where / = Diag(l, 1,1) and O is the zero matrix. For hyperelastic materials the scalar function (elastic potential) W : Q x M3 -> R+ exists such that E?(z,Q) = dW(x,Q)/dQ? for every Q 6 M3 and almost every x £ Q. If a material is incompressible, then det(Q) — 1, but for a compressible material |S(x,Q)| —> oo, W ( x , Q ) -> +00 as det(Q) ->• +0. We consider the following boundary-value problem. The quasi-static influences acting on the body are: a mass force with density / in £7, a surface force with density F on a portion F2 of the boundary, and a displacement v° of a portion F2 of the boundary is also given. Here F1 U F2 = dfl, F1 n F2 = 0 and area(F 1 ) > 0. For hyperelastic materials the finite elasticity BVP is formulated as the following variational problem (CIA 88, BRI 93-98)

where

Here V = {v : Q —> R3; v(x) = v ° ( x ) , x G F1} is the set of admissible displacements, (*,t>) is the specific and A(v) is the full work of the outside forces under the displacement v. It must be marked that even for "dead" forces, i.e. /, F = const(u,Vv), the specific work has the form (g,v)(x) = g a ( x ) v a ( x ) only in the Descartes coordinates (CIA 88, BRI 97). According to the general theory (EKE 76, FUC 80, TEM 83, GIU 84), for potentials of linear growth the set of kinematically admissible displacements is a subset of non-reflexive Sobolev's space Wlil(^l,R3)

We remind the definition of the limited static load (BRI 93). For this reason we introduce the set of addmissible "dead" outside forces for which the functional I ( v ) is bounded from below on V and, therefore, a solution of the problem (I) exists

This set is non-empty because for small outside influences the problem (1) is transformed into the classical variational problem of linear elasticity (LUR 90) which always has a solution (CIA 88; BRI 96a, 97).

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Definition 1. For outside forces (f,F) € B we examine the sequence of forces which are proportional to the real parameter t > 0. The number /* > 0 is named the limited parameter of loading and t*(/, F) is named the limited static load, if t(f, F] 6 B for 0 < t < t* and t(f, F) £ B for t > t*. The limited analysis problem is the investigation of the set of positive parameters t, for which the functional

is bounded from below on the set of admissible displacements (2). In practice the estimation from below for the limited static load is interesting because this information is sufficient for estimation of the strength of nonlinear elastic solid. Statement 2. For the limited parameter of loading the following estimation from below is true

where

According to the sense,

3. Discontinuous FEA and the partial relaxation From the previous author's results (BRI 96-98) it follows that for elastic potentials of linear growth the appropriate limited analysis problems need a relaxation (TEM 83). For variational problems with the multiple integral functional of linear growth the expanded space equals the BD space of vector-functions with bounded variations and generalized derivatives as the bounded Radon's measures (TEM 83, GIU 84). In the numerical analysis only finite dimension subspace of BD is used. Therefore, for the limited analysis problem (3) we will use the partial relaxation which is based on the special FEA with functions having breaks of the sliding type along ribs of simpleces (REP 89, BRI 98). Here we examine the plane limited analysis problem. Let Q C R2 and Q/j = UTh such that area(^\f7^) ->• 0 and length(dQ\<9Q/z) -)• 0 as h -)• +0, where

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T/j is the triangle and h is the characteristic step of the regular approximation (CIA 80). Every FEA is described by the set of nodes {xa}™=i and the set of ribs Gh = {^ap = [xa, £^]} including inside ribs and ribs on a portion F^ of the boundary dQhFor the displacement the following spacial piecewise continuous approximation is used where a,/? = l , 2 , . . . , r a such that rap £ G^, Uap is the component of the displacement in the node xa which is perpendicular to the rib rap, $fap '• fi/i —>• R is the piecewise linear discontinuous function such that ^!ap(x^} = ^a-> (a,/?,7 = 1,2, ...,m) and tyap ^ fypa. The supp (Vap) = supp (^a) consists of two triangles having the rib rap as common. If a rib rap £ F^ then the supp (^fap] consists of the only triangle. In this case the subspace V C Wlll(£l, R2) is approximated by the subspace Vh C BD(£l, R2) which is isomorphous to R2M, where M is the number of ribs in the set GhThe described FEA possesses the following properties. The component of the displacement, which is perpendicular to an appropriate rib, is continuous; but the tangent projection on this rib has a finite break. As a result, we have the special FEA with functions having breaks of the sliding type along ribs of triangles. The relaxated problem for the limited analysis problem (3) has the following form where

Here indeces "+" and "-" correspond to the displacement and the function of saturation on the triangles T£ and T^ having the common rib rap, index r corresponds to the tangent projection of displacement on this rib, and for ribs on Tlh the outside displacement is fixed, for example, v~ — vQh. Functions (v%, //i, Fh) are the standard spacial piecewise linear continuous FEAs of outside influences. According to the properties of FEA (CIA 80) and the results of paper (REP 89) we have th \ t_ as h —> +0 regularly. From the computational point of view the functional in the problem (5) is singular because it has no the classical derivative. Therefore, in this problem we use the simplest approximation of the modulus \z\ «3 (z2 + £2) regularization parameter e

1 / *?

with the

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It was proved that for potentials of linear growth the FE approximating algebraic systems can be badly determined (BRI 96c-98). As a result, for the numerical solution the decomposition method of adaptive block relaxation is used, because it practically disregards the condition number of the global stiffness matrix. The main idea of this method consists of iterative improvement of zones with "proportional" deformation by special decomposition of variables and separate calculation on these variables. 4. Numerical results

In the numerical experiments, the following boundary value problem was considered: a finite round rod is axial symmetrically stretched in the test machine by a given axial force P. In this case, the map is described by the following relation in the reference cylindrical coordinates

where p £ [0,1],

where fj, > 0 is the shear modulus under small deformations. For the limited stretching force P* the estimation from below P+ > \/3fjiTra2t- is true, where the parameter of loading t- is the solution of the following limited analysis problem

where

According to the convexity of domain, axial symmetry of the problem (6) and continuity of the axial component of dispalcement, the minimizer may have a break of the sliding type along the only line z — 1. This break is defined

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by a finite break of the function r(p, 1). Therefore, in the set V\ the condition r ( p , 1) = 0 is ignored, but in the functional the appropriate penal item is used

In the computational experiments the regular N x N triangulation of the domain (0, 1) x (0, 1) and the regularization parameter £ = 10"1 were used. In Figure 1 the experimental relations 77 *-+ tsh are shown. Lines 1, 2 and 3 correspond to the continuous FEA with N = 10, N = 20 and N = 40, respectively. Line 4 corresponds to the discontinuous FEA with N — 10. It is easily seen that continuous solutions converge to the discontinuous solution with increase of domain's discretization. The decrease of the regularization parameter e until 10~3 practically does not improve either continuous or discontinuous solutions.

[BAR 76]

BARTENEV G.M., ZELENEV Yu.V., A course in the physics of polimers, Chemistry, 1976 (Russian).

[BRI 93]

BRIGADNOV I.A., On the existence of a limiting load in some problems of hyperelasticity. Mech. of Solids, N° 5, 1993, p. 46-51.

[BRI 96a]

BRIGADNOV I.A., Existence theorems for boundary value problems of hyperelasticity. Sbornik: Mathematics, Vol. 187(1), 1996, p. 1-14.

[BRI 96b]

BRIGADNOV I.A., On mathematical correctness of static boundary value problems for hyperelastic materials. Mech. of Solids, N° 6, 1996, p. 37-46.

[BRI 96c]

BRIGADNOV I.A., Numerical methods in non-linear elasticity. In: Numerical Methods in Engineer ing'96. Proc. 2nd ECCOMAS Conf. (1996), Wiley, p. 158-163

[BRI 97]

BRIGADNOV I.A., Mathematical Methods for Boundary Value Problems of Plasticity and Non-Linear Elasticity. D.Sci. Thesis, St. Petersburg State University, 1997 (Russian).

[BRI 98]

BRIGADNOV I.A., Discontinuous solutions and their finite element approximation in non-linear elasticity. In: ACOMEN'98 — Advanced Computational Methods in Engineering. Proc. 1st Int. Conf. ACOMEN'98 (1998), Shaker Publishing B.V., p. 141-148.

[CIA 80]

ClARLET PH.G., The Finite Element Method for Elliptic Problems, North-Holland Publ. Co., 1980.

[CIA 88]

ClARLET PH.G., Mathematical Elasticity. Vol.1: Dimensional Elasticity, North-Holland Publ. Co., 1988.

Three-

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[EKE 76]

EKELAND I., TEMAM R., Convex Analysis and Vocational Problems, North-Holland Publ. Co., 1976.

[FUG 80]

FUCIK S., KUFNER A., Nonlinear Differential Equations, Elsevier Sci. Publ. Co., 1980.

[GIU 84]

GlUSTI E., Minimal Surfaces and Functions of Bounded Variations, Birkhauser, 1984.

[GRE 75]

GREEN A.E., ZERNA W., Theoretical Elasticity, Oxford University Press, 1975.

[LUR 90]

LURIE A.I., Nonlinear Theory of Elasticity, North-Holland Publ. Co., 1990.

[REP 89]

REPIN S.I., A variational-difference method for solving problems with functionals of linear growth. U.S.S.R. Comput. Math, and Math. Phys., Vol. 29(5), 1989, p. 693-708.

[TEM 83]

TEMAM R., Problemes Mathematiques en Plasticite, GauthierVillars, 1983.

Figure 1. The experimental relations between the geometrical parameter and the limited parameter of loading for different FEAs.

Entropy consistent finite volume schemes for the thin film equation Giinther Grun and Martin Rumpf Universitdt Bonn, Institut fur Angewandte Mathematik ABSTRACT We present numerical schemes for fourth order degenerate parabolic equations that arise e.g. in lubrication theory for the time evolution of thin films of viscous fluids. It turns out that a finite volume ansatz is the right approach to gain estimates on energy and entropy of discrete solutions. The latter are the key estimates to ensure nonnegativity of discrete solutions in a natural way. Another important feature is the question of tracing the solution's free boundary efficiently. This is achieved by a timestep control that makes use of an explicit formula for the normal velocity of the free boundary. Finally, we present some recent numerical experiments which indicate that also for fourth order degenerate parabolic equations a waiting time phenomenon occurs. Key words: fourth order degenerate parabolic equations, free boundary problem, finite volumes, adaptivity in time

1. Introduction

In this contribution, we will present new numerical schemes of finite volume type for fourth order degenerate parabolic equations of the form

Equation (1) is obtained as lubrication limit from the Navier-Stokes equations and models the height of thin films of viscous liquids that - driven by surface tension - spread on plain, solid surfaces. Assuming a no-slip boundary condition at the bottom of the thin film, the mobility becomes M ( u ] := |w| 3 , whereas the assumption of various slip boundary conditions leads to mobilities of the form M ( u ] — ci\u\3 + C2\u\f3 with positive numbers Ci,c 2 and ft 6 (0,3). From the analytical point of view, this initial boundary value problem shows a rather peculiar behaviour:

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• in contrast to solutions to nondegenerate fourth order parabolic equations, initially nonnegative solutions to (1) preserve nonnegativity (cf. [BF90], [Gr95], and [DPGG98]), • if M(u) = un with n £ (0,3), solutions to (1) exist that exhibit for t > 0 a zero contact angle at the contact line between liquid, solid and gas (cf. [Ber96a]. [Ber96b], and [BDPGG98]). Moreover, this contact line propagates with finite speed, i.e. we are dealing with a free boundary problem. However, if initial data have a nonzero contact angle, the propagation speed may be singular for / — 0. • no maximum or comparison principles are known. Those aforementioned issues also mean a great challenge in designing efficient numerical tools. A natural approach to guarantee nonnegativity of discrete solutions 1 is to develop a numerical scheme that allows for discrete counterparts of the relevant estimates - i. e. the energy and entropy estimate - known from the continuous setting. In section 2, we will introduce an implicit finite volume scheme which gives the perfect framework to realize this concept. Having presented in section 3 the relevant a priori-estimates in order to obtain compactness of sequences of discrete solutions, we will show in section 4 that a certain kind of harmonic integral means is the right choice for an entropy consistent numerical flux that allows for nonnegative discrete solutions. In section 5 we will introduce our method of timestep control which is based on a new, explicit formula for the velocity of the free boundary. This allows a tracing of the free boundary reminiscent of the tracing of shocks in hyperbolic conservation laws. On the other hand, the very formula for the velocity of the free boundary suggests that for sufficiently smooth initial data a waiting time phenomenon occurs, i.e. there is a slight delay in the onset of spreading. This phenomenon is well known for solutions to second order degenerate parabolic equations, like the porous media equation. We will present numerical simulations which give strong evidence that it also happens in the case of fourth order degenerate equations. 2. Deriving the entropy consistent finite volume scheme

The two major classes of discretizations for evolution problems - finite volume and finite element schemes - have both significant advantages. Finite volume schemes very easily lead to conservative schemes and incorporate fluxes on cell faces in a natural way, whereas finite element schemes correspond to a Galerkin discretization of the continuous problem and therefore carry strong provisions concerning a convergence analysis. In general, it is unusual to apply finite volume schemes to fourth order parabolic problems. But due to the peculiar diffusive structure of the elliptic term in (1), the thin film equation plays an

1

For a different ansatz to ensure nonnegativity , based on variational inequalities, we refer to [BBG].

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exceptionary role - as do other related higher order equations like the CahnHilliard equation with degenerate mobility, too. A suitable mixed finite volume - finite element discretization is the starting point to derive a conservative and entropy consistent numerical flux with an intuitive interpretation of the construction. In particular, the entropy consistency guarantees nonnegativity of the resulting numerical solution independent of the spatial resolution. Later on we will derive from the original finite volume scheme a finite element scheme, which will turn out to be preferable concerning further investigation in the numerical analysis. For simplicity we assume Q to be an interval in ID or a polygonally bounded domain in 2D, respectively. We suppose Q to be subdivided into cells. On these cells we suppose the discrete height U to be piecewise constant. We denote the negative Laplacian of the height which physically has the interpretation of pressure by p = —An. Finite elements allow a straightforward discretization of this Laplacian. Thus we have to find a suitable finite element mesh and function space for the pressure. We choose linear finite elements on the mesh dual to the finite volume mesh. To be more precise, we start with a simplicial grid Th on Q consisting of subintervals, respectively triangles E, on which the discrete pressure P will be defined as a function in the corresponding linear finite element space Vh, where h indicates the chosen grid size. Then a dual mesh is built of open dual cells Dx, again intervals, respectively polygonally bounded cells, corresponding to the vertices x of the primal mesh (cf. Figurel). We define a single dual cell by In the following, discrete functions will be denoted by uppercase letters, in constrast to lowercase letters for arbitrary functions in the nondiscrete function spaces. The discrete height U will be defined spatially constant on these dual cells. Figure 1 shows an example of such dual triangulations. To start with the

FIGURE 1. A 2D finite element triangulation whose edges are outlined in black and the corresponding dual finite volume mesh indicated by dashed lines. discussion of finite volume schemes, let us consider a cell D of the dual grid. On this subvolume we can rewrite equation (1) in conservation form

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where p — —Aw and v is the outer normal on 3D. The right hand side describes the inflow at the boundary, and A4(u)Vp is the corresponding flux. Thus, a numerical mobility M and a numerical pressure gradient are the main ingredients of a spatial discretization. As already mentioned, with the dual grid at hand the latter requirement is easy to fulfill. For given U we define P = —A/it/ on Vh, i.e. P is the unique function in Vh with

where J/, : C°(Q) —>• Vh is the nodal projection operator and (•, -}h indicates the well-known lumped mass scalar product corresponding to the integration formula (0, ^!}h := J fi Z/ l (0 1 3>). Gradients of P are by construction piecewise constant on elements E and thus almost everywhere on faces F of dual cells D. To pay account to the fact that the values U^ = lime_).o U(x±€v) may be different due to the discontinuity of U across cell boundaries, we suppose the discrete mobility M to be a function M : IR2 -> IR; (U+,U~) *-> M(U+,U~), where t/+, U~ are the outer, respectively interior values of U at the corresponding face. Finally, we can formulate our semi-discrete finite volume scheme

where Q(U+,U , VP) = M(U+,U )VP is the corresponding numerical flux. In our case we suppose that the discrete mobility M(U+,U~) is a symmetric matrix in IR x which is positive semidefinite and piecewise constant on E € ThThe resulting scheme is known to be conservative [Kr97] if this flux is symmetric, i.e. g(t/+,[/-, VP) = Q(U~,U+, VP) respectively

Thus, the inflow on F corresponding to D should coincide with the outflow with respect to the adjacent element at the face F. This immediately implies the conservation of mass f^Udx. Furthermore the flux should be consistent with the continuous flux q = A4 (w)Vp, i. e.

where the second term on the right hand side vanishes for decreasing grid size. There is still a great flexibility in selecting a numerical mobility. Let us recall that in case of hyperbolic conservation laws upwind discretization and entropy consistency conditions on the numerical fluxes, i. e. certain monotonicity properties, select the right entropy solution and guarantee moreover that neither artificial oscillations nor nonphysical shocks occur. These ideas carry over to the discrete modelling of thin films. For the trivial choice M(t/+, U~):=M.(U \u )Id nonnegativity of the numerical solution can no longer be guaranteed. Entropy consistency will ensure discrete nonnegativity independent of the selected grid size. In fact, we will be lead to some type of harmonic integral mean as an appropriate choice. This can

also be interpreted as a suitable type of upwinding. In the continuous setting an entropy is defined by

Choosing its derivative as a test function in the continuous problem we find that Jn G(u] is decreasing with time. This is in analogy to hyperbolic problems, where entropy estimates can be derived by testing the viscous approximated problem accordingly. These entropy estimates carry over to the discrete case provided we define otherwise. This numerical mobility can be regarded as a function M(XhU] on the primal grid. For the generalization to arbitrary dimensions, we refer to Section 4. Finally, the semidiscrete scheme can be discretized in time implicitly or explicitly. Therefore suppose [0, T] to be subdivided in intervals //- = (tk,tk+i] with tk+i — tk + Tk for time increments 77. > 0 and k = 0, • • • ,N — I. We will use backward difference quotients with respect to time which we shall henceforward denote by d~, respectively. Because of the significant stiffness of our problem we choose an implicit discretization. Otherwise a CFL-type condition r < C h4 would entail very small timesteps. In Section 5, we will discuss the selection of appropriate timesteps in detail. These ideas to construct entropy consistent finite volume schemes can be carried over to an appropriate pure finite element discretization. Therefore, we consider P and U both as functions in Vh and obtain the following finite element formulation of equation (1) with fully implicit, backward Euler discretization in time: For given U° E Vh find a sequence (Uk, Pk] for k = 0, • • • , N-1 Vh such that

for all system Lh the matrix

w'ithUk,Pk E

0, ty E Vh. Thus, a solution of (3) is obtained solving a nonlinear of q = dimVh equations for each time step. Let us define by Mft, standard lumped mass, respectively stiffness matrix and by Lh(W) the corresponding to the degenerate quadratic form, i. e.

Here we denote the nodal value vector for a function V E Vh by V , and with a slight misuse of notation rewrite Lh(W] for Lh(W). Then for given Uk £ IR9 we search Uk+1 E IR9 such that F(Uk+l) = 0 for

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Due to the absence of Dirichlet boundary conditions, A/> is not injective, i. e. kerA/j = {x H-> C\C (E IR) . This corresponds to the observation that Jn A/j£7 = 0. In contrast to the finite volume interpretation, the replacement of the exact mobility M ( - } by a certain quantity M(U) may now be interpreted as the choice of a specific quadrature to integrate the elliptic term numerically. For a certain class of grids in 2D (cf. [GR98]), a tedious but straightforward computation proves the equivalence of the finite volume and finite element approaches. 3. Existence, stability and compactness of discrete solutions

In this section, we recall the main results concerning existence and regularity. The proofs can be found in [GR98]. The key estimate for numerical analysis is the following energy estimate:

In ID a direct consequence of this estimate is a result on Holder continuity w.r.t. time for spatial averages of discrete solutions U which can be combined with the energy estimate to yield the following pointwise Holder regularity: Lemma 3.1: Assume d = 1 and that for integer I, k > 0 with I + k < N the relation kr > h4 holds. Then for a discrete solution (UTh,PTh) with \\M(t/r/OHoo < MI independently of r, h, there exists a constant C depending only on H^/ 0 ^ such that

for

As a consequence, convergence of discrete solutions to a solution in the continuous setting can be proven. For the quite different techniques to be used in higher space dimensions, we once more refer to [GR98]. 4. Entropy estimates — discrete nonnegativity in arbitrary space dimensions

In the case of space dimensions d > 1, the discrete mobility can no longer be given by a scalar valued quantity. This is due to the observation that on each cell of the dual mesh numerical fluxes coming from different directions have to be treated differently. It turns out that the right approach is to choose the mobility as a field of elementwise constant, symmetric positive semidefinite

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d x d-matrices which depends continuously on the discrete function U € Vh. To make the mobility matrix consistent with the entropy function, an additional admissibilitv condition has to be satisfied: Here, m is an appropriate approximation for the continuous mobility M. (for its explicit form depending on the smoothness of A4, we refer to [GR98]). Note that G is nonnegative and convex by construction. For nondegenerate reference simplices E(aij... iQd ):=convex hull(0, a\e\, • • • , Qd^d] we verify immediately that

satisfies the axioms above. For U(ctkek] — ^(0) the definition simplifies to Mkk = r n ( U ( Q ) ) . For elements E which can be mapped onto a reference element E by a rigid transformation x H-> x = XQ + A~lx, A an orthogonal matrix, the matrix M := AMA~l satisfies conditions ( i i ) , (iii). Since A is orthogonal, M is symmetric and positive semidefinite; hence condition (iv) is satisfied, too. For the general case, we refer to [GR98]. This construction allows to obtain the following discrete analogue of the entropy estimate: Lemma 4.1: Let (U, P) be a solution to the system of equations (3)-(4), and assume that (M, G) is an admissible entropy-mobility pair as described above. Then, for arbitrary T = Kr, K G IN, the following estimate holds:

As a consequence, the following result on nonnegativity of discrete solutions can be obtained which is in fine accordance with related results in the continuous setting (cf. [BF90] and [Gr95]): Theorem 4.2: (Existence of nonnegative discrete solutions U°h] Let Th be an admissible triangulation of Q and let n > 0 be the growth coefficient of M. in zero. Assume that the mobility M is monotoneously increasing and vanishes on IR_ U {0}. For arbitrary E > 0, there exists a positive control parameter O~Q which only depends on d, n, £, h and the initial datum UQ > 0

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such that: For every 0 < cr <

For a proof of this theorem, we refer to [GR98]. Let us remark that L. Zhornitskaya and A.L. Bertozzi [ZB] who studied finite difference schemes for growth coefficients n > 2 obtained quite similar results on positivity of discrete solutions.

FIGURE 2. Numerical approximation of the solution to ut + div(« • VAw) = 0 for initial data given as the characteristic function of a nonconvex set 5. Implementation, timestep control, waiting time phenomenon

One of the most important questions with respect to numerical simulations of wetting phenomena is how to trace the solution's free boundary in an efficient way. In order to describe the arising difficulties, let us first consider questions of implementation. In each timestep, we have to solve a nonlinear system (cf. section 2). In fact, we first consider a related semi-implicit system, given by For the solution of the fully implicit scheme, we apply an appropriate fixedpoint iteration to satisfy the original problem with W = U (for details cf. [GR98]). Now observing that in the semi-implicit scheme the numerical free boundary

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cannot propagate more than a distance h in each timestep, it is reasonable to choose the time increment r smaller than the quotient e^d(t] wnere speed(t] stands for the maximum normal velocity of the numerical free boundary at time t. As a consequence of this special choice of time increment, only a very small number of iterations is necessary to obtain the solution of the fully implicit scheme. Formal considerations - performed in the continuous setting - indicate that the normal velocity Vn(^(i]} of the free boundary in a point £(t) can be related to spatial derivatives of u in £(t) according to the following formula:

This formula has been proved for self-similar source-type solutions in [GR98]. In the framework of the algorithm studied in this paper, we formulate a discrete counterpart of formula (7) in the following way: In a timestep ^, we first determine on each E £ Th numbers otherwise.

Then, we define the time increment by the formula

If 77, > 1, the results on Holder continuity in space for discrete solutions allow to give a robust, but coarse upper bound: This implies for the time increment: Hence, the assumption r > /i4 in Lemma 3.1 does not mean any restriction any longer. Formula (7) indicates that for sufficiently smooth initial data the velocity of the free boundary vanishes. So let us take Q — (0, 1) and as initial data the function UQ(X) = f [cos (|TT^)] 1 . We choose M(u] = u2 in equation (1) and obtain for / 6 [0,1] a solution u as shown in the six diagrams on the left of figure 3. From top left linewise to bottom right, they represent six snapshots of u(t, •) for increasing times t. To have a closer look at the behaviour at the free boundary for small £, we depict on the right the function otherwise at four different times ranging from t — 0.0 in the background to t — 2.5 • 10 2 in the foreground. It turns out that for t < 2.6 • 10~ 2 , the free boundary does not move, whereas for larger times the support monotoneously increases. This

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FIGURE 3. Delayed onset of spreading for solutions to the equation ut + (u2 • uxxx}x — 0 and sufficiently smooth initial data(number of gridpoints: 500) gives very strong evidence that also for the thin film equation a waiting time phenomenon occurs. For other simulations illustrating the variety of phenomena encountered in thin film flows, we refer the reader to [GR98] and [Gr99].

REFERENCES [BBG]

J. Barrett, J. Blowey, and H. Garcke. Finite element approximation of a fourth order nonlinear degenerate parabolic equation, to appear in Numer. Mathematik. [BDPGG98] M. Bertsch, R. Dal Passo, H. Garcke, and G. Griin. The thin viscous flow equation in higher space dimensions. Adv. Diff. Equ., 3:417-440, 1998. [Ber96a] F. Bernis. Finite speed of propagation and continuity of the interface for thin viscous flows. Adv. in Diff. Equations, 1, no. 3:337-368, 1996. F. Bernis. Finite speed of propagation for thin viscous flows when 2 < n < 3. [Ber96b]

C.R. Acad. Set. Parts; Ser.I Math., 322, 1996. [BF90] [DPGG98]

[Gr95] [GR98] [Gr99]

F. Bernis and A. Friedman. Higher order nonlinear degenerate parabolic equations. J. Diff. Equ., 83:179-206, 1990. R. Dal Passo, H. Garcke, and G. Griin. On a fourth order degenerate parabolic equation: global entropy estimates and qualitative behaviour of solutions. SIAM J. Math. Anal, 29, 1998. G. Griin. Degenerate parabolic equations of fourth order and a plasticity model with nonlocal hardening. Z. Anal. Anwendungen, 14:541-573, 1995. G. Griin and M. Rumpf. Nonnegativity preserving convergent schemes for the thin film equation. 1998. Preprint No. 569, SFB 256 University of Bonn. G. Griin. On the numerical simulation of wetting phenomena. 1999. to appear in Proceedings of 15th GAMM-Workshop, Kiel.

[Kr97]

D. Kroner. Numerical Schemes for Conservation Laws. Wiley and Teubner,

[ZB]

Chichester and Stuttgart, 1997. L. Zhornitskaya and A.L. Bertozzi. Positivity preserving numerical schemes for lubrication-type equations. SIAM Num. Anal, submitted, 1998.

Convergence of finite volume methods on general meshes for non smooth solution of elliptic problems with cracks Philippe Angot Universite de la Mediterranee, I.R.P.H.E. Chateau-Gombert, 38 rue F. Joliot Curie, F-13451 Marseille Cedex 20 - E-mail : [email protected]

Thierry Gallouet and Raphaele Herbin Universite de Provence, C.M.I. - L.A.T.P., 39 rue F. Joliot Curie, F-13453 Marseille Cedex 20 - E-mail : [gallouet,[email protected]

ABSTRACT A model of insulating cracks for elliptic problems is presented and proved to be well-posed. The solution is indeed discontinuous. A finite volume scheme on general polygonal meshes is introduced to solve such problems. Since no unknown is required at the fracture interface, the scheme is as cheap as more standard schemes for the same problems without cracks. With weak regularity assumptions, we establish for discrete norms some error estimates in O(h] where h is the maximum diameter of the control volumes of the mesh. Key Words: elliptic problems, insulating cracks, discontinuous solutions, finite volumes, fracture resistances, error estimates.

1. Introduction The concept of contact resistance is sometimes introduced empirically for diffusion problems (Pick's law) with imperfect contact, e.g. thermal (Fourier's law) or electrical (Ohm's law) contact resistance, or also hydraulic resistance of fissure for flows in porous media described by the Darcy's law. The objective is to take account of fault lines or too thin layers compared to the largest scale under study. Hence, from this macroscopic scale the solution at the interface is indeed discontinuous. In previous works e.g. [ANG 89], we generalized this concept and formulated for such elliptic problems a mathematical model with discontinuous coefficients which includes a jump transmission condition linking the divergential flux with

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the jump of the solution at the interface. It ensures the well-posedness of the associated elliptic or parabolic problems and we proved in [ANG 98, ANG 98b] the global solvability within a variational framework. Besides, we have shown how to use the imperfect transmission problem for fictitious domain modelling with immersed boundary conditions imposed by a penalty method. In that case, we performed the convergence analysis and derived the associated error estimates as functions of the penalty parameter; see also [ANG 97]. For the numerical solution on a rectangular mesh, we proposed in [ANG 89, ANG 89b] an original finite volume method, either cell-centered or vertex-centered, and based on the introduction of "fracture resistances" at the faces of the control volumes. For this scheme, some error estimates in O(h) are established in [ANG 98, ANG 98b] and various numerical results have illustrated the capabilities and the efficiency of this methodology. In the present work, we generalize the model in two ways; see also [ANG 99]. First, we do the mathematical analysis in the case where the fracture interface £ is an open surface strictly included in the bounded domain fi, e.g. without any connection with its boundary F = d£l. We prove the existence and uniqueness of the solution for a diffusion-reaction problem. This case is more difficult because the open domain is no longer located locally on one side of its boundary since the fault interface does not divide fi into two disjointed subdomains. Second we extend, in the case of a polygonal interface, the finite volume scheme to general polygonal meshes, as considered in [EYM 97] or [HEI 87, SHA 96], e.g. triangular [HER 95] or Voronoi meshes; see also [COU 96, HER 96, LAZ 96]. The construction of a general admissible mesh is made in such a way that the discontinuity lines of the operator coefficients, and/or the polygonal fracture interface lie on faces of some control volumes. Then, we construct a finite volume scheme including fracture resistances at faces of control volumes, well-suited to the numerical approximation of the imperfect transmission problem. We show how to satisfy both conservativity and consistency of the numerical fluxes. Indeed, the numerical scheme is locally conservative by construction. Let us notice also that our numerical scheme inherently involves the locally conservative approximation of the immersed jump condition (3) without using unknowns located on the interface £. Hence, it only uses a four-point stencil for triangular finite volumes in 2-D, or a six-point one in 3-D. This means that the solution cost is as cheap as for a more classical finite volume scheme without any fracture interface, and hence cheaper than with the "double-node" finite difference scheme proposed in [SAM 78]. 2. Well-posed elliptic model for insulating cracks

Let the domain fr C Md (d — 2 or 3 in practice) be an open bounded polygonal set, T = d£l being its boundary, which includes a polygonal interface £ C Md~l. Let us define the open bounded set fi such that fi = fi U £ and its boundary F = dft = f U £. It is always possible to prolong £ within a

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polygonal interface £ D £ which divides the domain fi into two disjointed subdomains fi~ and fi+ such that & — fJ~ U £ u n + . We denote by x~ and X+ the characteristic functions of f)~ and fi+, respectively. Let n be the normal unit vector on £ oriented from fl~ to f) + . For the data / £ L 2 (f2), we consider the second-order elliptic problem for the real-valued function u defined in £) :

where the symmetric second-order tensor of diffusion a = (a-ij}\*
For the sake of clarity, we restrict ourselves to homogeneous Dirichlet boundary conditions on F, although other kinds of boundary conditions can be considered by using the results in [EYM 97], as well as more general or nonlinear elliptic or parabolic problems. In the usual transmission problem, the perfect transmission conditions on £ are considered, i.e. continuity of the traces of both the solution u\x and normal component of the flux vector (u)- n|s = —(a- Vw)- n|s on £, and the solution of the perfect transmission problem is the unique solution u 6 HQ(&) satisfying (1) over the whole domain J7. 2.1. Jump boundary condition at the interface We now investigate the case of an imperfect contact on £ with the model for "insulating cracks", see [ANG 98]. We consider the following jump or fracture boundary condition on £ which assumes the normal diffusive flux *

where [u]s denotes the jump of the solution on S oriented by n, g is given in Z/ 2 (S) and (3 is the transfer coefficient on £ satisfying the following assumption:
The inverse of ft can be defined as the fracture resistance p = l / f t through the interface £. When ft = 0 on £, the condition (3) degenerates into the non-homogeneous Neumann boundary condition on £: (p(u)-n\^ = g. The
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condition (3) written with p instead of /3 yields the perfect transmission condition when p = 0 on S, see the proof in [ANG 98b] and also Sec. 3.1. A particular case of the condition (3) when g = 0 is also considered in [SAM 78]. 2.2. Global solvability of the imperfect transmission problem We study within a variational framework the global solvability of the previous problem (1-3) with the assumptions ( A l ) , (A2), (-43), g e £ 2 (£) and / 6 L 2 (ft). Obviously, the solution u £ HQ(&) when [wjs ^ 0. We begin by some technical results for the traces of functions of Hl (ft) on <9ft = f U S. Lemma 1 (Traces on T and S in Hl(Q)) If ft is an open bounded polygonal subset of lRd including a polygonal surface S C IR ~ such that $1 — ft U S and dft = <9ft U S, hereafter called the configuration hypothesis (H), then we can define the following trace applications 7, 7+, 7" for all v in Hl(fl) by: 7(v) = v \ f , 7 + (f) = v + |£, 7~~(v) = v~\-£. They are linear and continuous from Hl(ty intoL2(f) orL 2 (S) respectively, i.e. 3c(ft), c+(ft), c~(ft) > 0 such that V v £ H l ( f t ) : |Hf|| L 2 ( f) < c(ft)|H| H i ( n ) , ||^ + |E||L 2 (E) < c + (n)||v|| f / i { n) and l|v~|s|U2(E)

and 7~(t») = (7o^|n-)|s- The key point is to show that these trace operators are independent on the choice of fi+ or fi~; This can be proved at least for a polygonal boundary of ft, i.e. if (H) holds; see [ANG 99]. D Lemma 2 (Jump of traces on S in Hl(Q)) If (H] holds, then : PROOF. We get for all v G Hl(Q), using the continuity of traces in Lemma 1 We now define the Hilbert space W = {v € L 2 (ft) n Hl(ty; 7(7;) = u|f = 0} equipped with the usual inner product and associated norm in H l ( f y . Then we prove the following proposition by considering the variational formulation of the problem (1-3) in ft as below : Find u G W such that Vv G W,

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Theorem 1 (Existence and uniqueness for the crack model) // the assumptions (H), (Al), (A2) and (A3) hold, the problem (1-3) with g 6 £ 2 (£) and f £ L 2 (Q) has a unique weak solution u £ W satisfying (4) Vu £ W, such that:

PROOF. The bilinear symmetric form in the left-hand side of (4) is clearly coercive in W. By using Lemma 2, we easily show the continuity of both this bilinear form in W x W and the linear form, i.e. the right-hand side of (4), in W. Then we apply the Lax-Milgram theorem which completes the proof. D 3. Finite volume approximation on general meshes For the sake of simplicity, we only consider here the case of an isotropic tensor of diffusion, i.e. a scalar coefficient variable in space a(x) satisfying (Al). The general case of a diffusion matrix a(x) requires an additional condition of quasi-regularity of the mesh, e.g. [EYM 97]. Definition 1 (Admissible meshes) Let 17 = 17 U £ be an open bounded polygonal subset of IRd, d — 2, or 3 satisfying (H) in Lemma 1. An admissible finite volume mesh of 17; denoted by T, is given by a family of "control volumes", which are open polygonal convex subsets of 17 , a family of subsets of 17 contained in hyperplanes of JRd, denoted by 8 (these are the edges (twodimensional) or sides (three-dimensional) of the control volumes), with strictly positive (d — I)-dimensional measure, and a family of points of 17 denoted by P satisfying the following properties (in fact, we shall denote, somewhat incorrectly, by T the family of control volumes): (i) The closure of the union of all the control volumes is 17. (ii) For any K £ T, there exists a subset £K of 8 such that OK = K \ K = U0. e< r K cr. Furthermore, £ = UK^T^-K(Hi) For any (K, L) £ T2 with K ^ L, either the (d— I)-dimensional Lebesgue measure of K Pi L is 0 or K fl L — W for some a £ £, which will then be denoted by K\L. (iv) The family P = (XK)K^T is such that XK £ K (for all K £ T) and, if a ~ K\L, it is assumed that XK ^ XL, and that the straight line T*K,L going through XK and XL is orthogonal to K\L. (v) There exists £^ C. £ such that E = U ff ££ s <7. (vi) For any a £ £ such that a C 5H = d$l U E, let K be the control volume such that a G £K- If XK £ ai ^ ^K,a be the straight line going through XK and orthogonal to a, then the condition T>K,a H a ^ 0 is assumed; let y

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In the sequel, the following notations are used. The mesh size is defined by: hj- = sup{diam(K), K € T}. For any K 6 T and a 6 £, m(K) is the ddimensional Lebesgue measure of K (it is the area of K in the two-dimensional case and the volume in the three-dimensional case) and m(cr) the (d — \)dimensional measure of a. The set of interior (resp. boundary) edges is denoted by £int (resp. £exij, that is £int — [a G £; a $_ dfr} (resp. £ext = {cr e £; o C d$i}). The set of neighbours of K is denoted by N(K], that is J\f(K) = [L £ T; 3cr G £K, o" — K r\ L}. If o~ = K\L, we denote by da or d^.\L the Euclidean distance between XK and XL (which is positive) and by G?K,

and *< = ^K) IKb(x)dx' fK = ^K) fKf(x">dx' P* = ^7) L P(X"> ds(x^ 9* ~ ^y f a 9 ( x ) d s ( x ) , or simply by aK = a(xK], bK = b(xK), fK = /(XK}, Pa = p(y<j} or ga = g(ya} if the coefficients are regular enough to define these quantities. Define X(T} as the set of functions from fi to IR which are constant over each control volume of the mesh. For v € X(T] such that VK denotes the value taken by v on any control volume K € T, define the discrete HQ(&) norm by :

where

and

3.1. Finite volume method with fracture resistances As in [ANG 89, ANG 98] for a rectangular mesh, the scheme is derived with respect to p|s instead of (3\% because it is more practical since the case when p\Y, = 0 and g\

For a £ £ e xt> the Dirichlet boundary condition is written. Finally, the numerical scheme reads in the following synthetic form: for all K £ T,

3.2. Convergence analysis and error estimate Theorem 2 (Error estimate of the FV scheme) // ("H) and the assumptions (Al), (-42), (.A3) hold, let u £ W be the unique weak solution of the problem (1-3) with g £ L 2 (£), / £ L 2 (f)). Let T be an admissible mesh for the discretization of that problem, in the sense of Definition 1. Then, there exists a unique discrete solution (UK)K^T to the finite volume scheme (6-7-8-9-10-11). Moreover, assume piecewise sufficiently regular data: \/K £ T', f\K £ C Q ( K ] , b\K € Cl(K) and O\K in Cl(K], and Vcr £ £•%, P\a o,nd g\a in Cl(a], and such that the solution u £ Hl($l) satisfies U\K £ C2(K], for any K £ T. Define the error ej- £ X(T) by e-r(x] = U(XK) — UK for a.e. x £ K, and for all K £ T. Then, there exists C > 0 only depending on u, a, b, p and SI such that the following error estimates hold for the discrete HQ and L2 norms: and SKETCH OF PROOF. It is inspired from [EYM 97]. With the conservativity property, a discrete coercivity lemma is proved which ensures both the existence and uniqueness of the discrete solution via a discrete maximum principle and the stability in the sense of finite differences. By using Taylor expansions and the regularity assumptions, the weak consistency in O(h-j-) of the approximate volumic terms and the numerical fluxes, specially for (3) on S, is proved. Then it yields the error estimate for the discrete HQ norm. Finally, the L2 estimate follows from a discrete Poincare inequality. D Remark 1 (Convergence and weaker assumptions) With no more regularity on the solution than u £ Hl(Q), it is possible to prove directly the convergence of the scheme by using some compactness results and a discrete trace

222

Finite volumes for complex applications

lemma established in [EYM 97]. Besides, error estimates in O(hj-) like above, also hold for weaker regularity assumptions on the solution, i.e. u G Hl(£l] and U\K £ H^(K) for any K £ T. It requires the use of Taylor expansions with integral residuals to prove the weak consistency. See [ANG 99] for the details. References [ANG 89] [ANG 89b]

[ANG 97]

[ANG 98]

[ANG 98b]

[ANG 99] [COU 96]

[EYM 97]

[HEI 87] [HER 95]

[HER 96]

[LAZ 96]

[SAM 78] [SHA 96]

ANGOT PH., Ph.D. Thesis, University of Bordeaux I, april 1989. ANGOT PH., Modeling and visualization of thermal fields inside electronic systems under operating conditions, IBM Technical Report, TR-47095. 70 p., april 1989. ANGOT PH., Analysis of singular perturbations on the Brinkman problem for fictitious domain models of viscous flows, M2 AS Math. Meth. in the Appl. Sci., Preprint 1997 (to appear). ANGOT PH., Finite volume methods for non smooth solution of diffusion models; Application to imperfect contact problems, in Advances in Numerical Methods and Applications, O.P. Iliev et al. (Eds), Proceedings 4th Int. Conf. NMA '98, Sofia, World Scientific Pub., 1999. ANGOT PH., Mathematical and numerical modelling for a fictitious domain method with penalized immersed boundary conditions, Preprint HDR Thesis - Univ. Mediterranee, 1998 (submitted). ANGOT PH., GALLOUET TH., HEREIN R., in preparation. COUDIERE Y., VILA J.-P., VILLEDIEU P., Convergence of a finite volume scheme for a diffusion problem, in Finite Volumes for Complex Applications - Problems and Perspectives, F. Benkhaldoun and R. Vilsmeier (Eds), Hermes, p. 161-168, 1996. EYMARD R., GALLOUET TH., HEREIN R., Finite Volume Methods, in "Handbook of Numerical Analysis", P.G. Ciarlet and J.L. Lions (Eds), North-Holland, Preprint 1997 (to appear). HEINRICH B., Finite Difference Methods on Irregular Networks, Int. Series Numer. Math. 82, Birkhauser Verlag, Basel, 1987. HEREIN R., An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh, Num. Meth. for P.D.E., 11:165-173, 1995. HEREIN R., Finite volume methods for diffusion convection equations on general meshes, in Finite Volumes for Complex Applications - Problems and Perspectives, F. Benkhaldoun and R. Vilsmeier (Eds), Hermes, p. 153-160, 1996. LAZAROV R.D., MISHEV I.D., Finite volume methods for reactiondiffusion problems in Finite Volumes for Complex Applications Problems and Perspectives, F. Benkhaldoun and R. Vilsmeier (Eds), Hermes, p. 233-240, 1996. SAMARSKII A.A., ANDREEV V.B., Methodes aux Differences pour Equations Elliptiques, Editions MIR Moscou, 1978. SHASHKOV M., Conservative finite-difference methods on general grids, CRC Press New York, 1996.

Application and analysis of finite volume upwind stabilizations for the steady-state incompressible Navier-Stokes equations

Lutz Angermann Institut fiir Analysis und Numerik Fakultdt fur Mathematik Otto-von-Guericke-Universitdt Magdeburg PF 4120 D-39016 Magdeburg, Germany

ABSTRACT Among the various difficulties in the numerical solution of NavierStokes equations, the case of high Reynolds numbers requires extra care in the choice of a concrete discretization scheme. The specific treatment of the nonlinear connective term by means of finite-volume-based approaches within the framework of conforming or nonconforming finite element m.ethods forms one class of practically relevant stabilization techniques. The aim of the contribution is to describe a general approach to the design of those methods in multidimensional situations and to extract the basic principles of their analysis. Key Words: finite volume upwind method, incompressible Navier-Stokes equations

1. Introduction

Let c > 0 be a real number, Q C ffi d with d = 2 o r c / = 3 b e a bounded, polyhedral domain with Lipschitz continuous boundary and / : Q —>• Md be a given vector field. With V := W$(tt)d and Q := L*(tt) := {q € L2(Q) : (q, 1) = 0}, the following weak Navier-Stokes problem is considered: Find (ti, p) = (it 1 ,... , ud, p) € V x Q such that it holds:

224

Finite volumes for complex applications

where the trilinear form n : V3 —)• M is defined as n(w,u,v) := ((w • V)it,u) and (•, •) denotes the inner product of L^Q] or L2(Q)d• Now let Vh,Qh be members of two families of finite element spaces approximating, in certain sense, the elements of V, Q. Here, the discrete spaces need not be subspaces of V, Q. While it is not difficult to replace the forms (Vit, Vv) and (p, V • v) by their "broken" counterparts on the underlying partitions 7/i of Q

the trilinear form n has to be defined in a more careful way. In the book [RST 96, Sect. IV.2], the authors pointed out that the following three properties of the discrete form n/j : Vft3 —>• M are sufficient conditions to establish convergence of the numerical method: Semidefiniteness: Lipschitz-continuity: Consistency:

where || • ||/j is a norm on Vh and //, : W$(Q}d fl V —>• VA denotes some interpolation operator. The following general approach for the design of the discrete trilinear form satisfying the above conditions is based on three different partitions of Q. One of the partitions is denoted by Th and is either a triangulation (i.e. it consists of d-simplices) or a block-partition (i.e. it consists of convex quadrilaterals (d — 2) or convex hexahedra (d — 3)). It is assumed that Th is admissible in the usual sense, i.e. two elements of the partition are allowed to have in common either a vertex or an edge or, if d — 3, a face. Using the notation T for the elements of Th, the parameter h of the partition is defined as usual: If T has the diameter HT, then h := max h?- Notice that this partition is related T&TH to the approximation Uh of the unkonwn u; in certain situations the partition for the discrete unknown ph may differ from ThNext, on each partition Th a (not necessarily conforming) finite element space is defined, the elements of which are piecewise polynomially of maximal degree / (E N U J O } . In particular, the polynomial space on T may be incomplete, especially for quadrilateral/hexahedral elements. Given some finite element space, the corresponding set of functionals (global degrees of freedom) naturally splits into Langrangian functionals and others, where a Langrangian functional is defined via point values of its argument. Therefore, considering these Langrangian functionals, a collection of (global) nodes can be associated in a natural way. This collection of nodes can be subdivided into the class of nodes lying on element boundaries and the class of nodes belonging to the interior of some element. For example, the second

Numerical analysis

225

class is empty for all serendipity elements, but for the mini element it contains all element barycenters. Let A.g denote the set of indices of all the nodes from the first class. The subset A.g C A.g contains, by definition, the indices of interior (w.r.t. Q) nodes only. Finally, declare d\g := A f f \ A g and let \gT C As contain the (global) indices of the nodes belonging to the element T. Due to the boundary conditions, the above finite element space is restricted to elements satisfying a discrete boundary condition, i.e. we set

The distance between two nodes #,, Xj ( i , j £ A9) is denoted by dij. Now, an important observation is that the index set A.g can be decomposed into two disjoint subsets A, A such that A9 — A U A and A fl A =0. Such a decomposition can be generated, for example, by a hierarchical decomposition of the finite element space Shi into a "lower degree part" and its linear complement. Then A corresponds to the nodes of the first part and A to the complement's nodes. Obviously, this decomposition induces similar ones of A5, d\.g and A^, respectively. If Shi is a space of elements of low degree, then it is allowed that the decomposition is trivial, i.e. the complement may be the trivial space consisting of the zero element only. In this case, A is empty by definition. As an example, in the case when Shi is built by means of conforming /Velements (Taylor-Hood elements, d = 2), we have the decomposition

where Shi is the space of conforming Pi-elements (Courant elements) and S^ is the space of so-called "bump functions", i.e. the set of the piecewise quadratic nodal basis functions associated with the edge midpoints. To describe the discretization of the trilinear form n, a further partition of Q is needed. This second partition 7^* consists of subdomains Q2 C ^, the boundary part Q D d&i of which is the union of subsets of (d — l)-dimensional hyperplanes. An incidence relation between these two partition is defined by the help of the nodes of Th, i.e. each f22 should correspond to one node Xi and vice versa. Further, for all indices i, _;' G A, j ^ z, let F,j := Q, D fij and mzj := measrf_i ( F z j ) . Then it makes sense to introduce the index set

As a consequence of these definitions, the following representation of dQ,i is valid: d£li — M F,-j, i 6 A. Moreover, there are the obvious symmejeA,

try relations djj = dij, Fj,- = F,-j, ra^- = m,-j. The boundary parts F;J can be structured in a finer way: F^- := F,-j fl T for T € Th- Analogously, mj := meas d _i (F^) . Now we formulate a set of geometrical assumptions.

226

Finite volumes for complex applications

(Al) There exists a constant C > 0 such that, for all /, / £ A, it holds (A2) There exists a third partition {&?•}. . -r-. ,. „ T >0n of fi such that the l J J z , j € A,t7=j, T:m^ subdomains Q^ have the following properties:

(iii) Each ^ij- can be decomposed into a finite number ifj r of pairwise ™ i

T

i

f i

disjoint open c/-simplices Q-' such that Q 2 = U/=i ^W an-d, for any / £ [1,/^-JTv? ^,-/ is the image of a fixed (reference) simplex T under a regular affine transformation, where the pre-image F of F. ' := FJJnf2,-j' does not depend on the particular values of z, j, T, /. rp

,

T1^

(iv) On each F - • ' , the unit outer (w.r.t. Q z -) normal i/-•' is constant. T1

I

T~>

/

The diameter of the simplex 0,- •' is denoted by /i-•' . Furthermore, we set m-'T I := measd_i I( F -T- 'l j\ I. Finally, we define 17jj := int I (A3) There exists a constant C > 0 such that, for all z,j £ A, it holds

(A4) There exists a constant C > 0 such that, for all i, j £ A, T £ 7/j, / £ [I,/^-]AT, it holds mfy c?,-j < Cmeas^ fSl f y J . (A5) There exists a constant C > 0 such that, for all i,j £ A, T £ 7/i, / £ [1, /5]jv, it holds (/ij'')4 < Cmeasd (^'') . A dimensional analysis of the quantities appearing in the Assumptions (Al), (A3), (A5) easily shows that these conditions are not very restrictive.

2. Discretization (Treatment of the trilinear form n)

Because of

the description of the discretizaV

tion can be restricted to the scalar case. So let w £ Wl(£l}d be such that

Numerical analysis 227

V • w — 0 and define, for w, v £ W / 2 1 (fi), the form

It is not difficult to see that ns can be represented equivalently as

Taking into consideration the condition V • w •= 0, it is not so far to omit the last term, i.e. we get

where This is the starting point for the discretization. Suppose there is some control function r : M —> [0, 1] satisfying the following properties: • r(z) is isotone for all z,

zr(z) is Lipschitz-continuous on the whole real axis. /*

Furthermore, set 7,-; := m"31 / J

JrtJ

/ **V ' " ft ' ' \

v • w ds and ra := r(

\

IJ IJ

s

}. We mention

J

that 7jj is antisymmetric, i.e. 7^,- = — 7^-. Moreover, in the definition of 7;j, the value of Jr v • w ds can be replaced by certain approximation which has to satisfy, among natural error estimates, the above antisymmetry condition. Then, by standard arguments in the derivation of finite volume discretizations (cf. [ANG 95]), we can write

Thus we get

228

Finite volumes for complex applications the quantities

Now, redefining for as well as

we set

Returning to the original form n, we set for

COROLLARY 1 // the control function r satisfies (P5), then it holds

Typical examples of the control functions are

(full upwind

(exponentially fitted scheme),

scheme), (Samarskij's scheme).

Finally, some discrete forms and operators have to be introduced. For

we set where

The extension of these definitions to the case of Rd-valued functions is obvious and will not be denoted separately. By Ih '• W%(£1] —>• Shi, an interpolation operator is denoted, whereas Lh '• C(fi) —>• ico(^) stands for a so-called lumping procedure. That is, the image of Lh is the subspace consisting of functions being constant on the elements of the secondary partition 7^*. Concrete properties of these operators are collected in the subsequent assumptions.

3. Properties of the discrete forms In order to verify the required properties of Lipschitz-continuity and consistency of 77,5/2, we formulate further assumptions.

Numerical analysis 229

(A6) There exists a constant C > 0 independent of h such that, for all v/, G SM, it holds

(A7) For arbitrary p G [1,6], there exists a constant C > 0 such that, for all Vh € SMJ*> holds (A8) There exists a constant C > 0 such that, for arbitrary p G [1,6] and for all Vh G SM, it holds

Since A is non-empty, in general, ||^/jV/,||o,p,n is on ly a seminorm on Shi(A9) There exists a constant C > 0 such that for all v/, G Shi, it holds

(A10) There exists a constant C > 0 such that (i) for all v G W%(Q), it holds

(ii) for all v G W%(ti) and all T G T/,, it holds

(iii) for arbitrary p G (rf, 6], u G W 1 ^)

and all T G TA, it holds

(All) There exists a constant C > 0 such that (i) for arbitrary p G (d, 6], v G W^(n) and all T G TA, it holds:

(ii) for all v G W$(tt) and all T G Th, it holds

(A 12) There exists a constant C > 0 such that, for all v G W$(£l) and all T G T/M it holds

230

Finite volumes for complex applications

Now, the folllowing results can be proved (for details, see [ANG 98]). LEMMA 1 Suppose (Al), (A2), (A3), (A4), (AS), (A6), (A7), (A8). Then, for arbitrary Wh^Zh 6 Vh and u/^v/i G Shi the estimate

holds, where C > 0 is a constant which does not depend on h. LEMMA 2 Suppose (A2), (A4), (A6), (Al), (A8), (A9), (AW), (All), (A 12). Then, for any w £ Vl / 2 2 (^) rf fl V satisfying V • w = 0, any u £ 0

W-2(£l] n W£(£l} and any element Vh € Shi the estimate

holds, where C > 0 is a constant which does not depend on h.

4. Application The above approach can be used to give an alternative proof of the convergence properties of Schieweck's family of nonconformingquadrilateral/hexahedral elements [SCH 97] which find successful application in parallel Navier-Stokes codes. Details for the case of the so-called Pi-parametric element are described in [ANG 98]. 5. Bibliography [ANG 95]

ANGERMANN, L. Error estimates for the finite-element solution of an elliptic singularly perturbed problem. IMA J. Numer. Anal, 15, 1995, p.161-196.

[ANG 98]

ANGERMANN, L. Error analysis of upwind-discretizations for the steady-state incompressible Navier-Stokes equations. Fakultat fur Mathematik, Otto-von-Guericke-Universitat Magdeburg, Preprint Nr. 33, 1998.

[RST 96]

Roos, H.-G., STYNES, M. AND TOBISKA, L. Numerical methods for singularly perturbed differential equations. Springer- Verlag, Berlin-Heidelberg-New York, 1996.

[SCH 97]

SCHIEWECK, F. Parallele Losung der stationaren inkompressiblen Navier-Stokes Gleichungen. Habilitationsschrift, Fakultat fur Mathematik, Otto-von-Guericke-Universitat Magdeburg, 1997.

A new cement to glue non-conforming grids with Robin interface conditions

Yves Achdou Insa Rennes, 20 Av. des Buttes de Coesmes 35043 Rennes, France

Caroline Japhet, Frederic Nataf CMAP, Ecole Polytechnique 91128 Palaiseau, France

Yvon Maday Laboratoire d'Analyse Numerique, Universite Pierre et Marie Curie 4, place Jussieu, 75252 Paris Cedex 05, France

ABSTRACT We propose and analyse a domain decomposition method based on Schwarz type algorithms that allows for an extension to optimized interface conditions on nonconforming grids. We consider the convection-diffusion equation discretized by a finite volume method. The nonconforming domain decomposition method is proved to be well-posed and the error analysis is performed. Then we present numerical results that illustrate the method. Key Words: Domain decomposition methods, optimized artificial interface conditions, non-conforming grids, convection-diffusion problems, finite volume methods, parallel computing, High Performance Computing.

1. Introduction The goal of our project is to design domain decomposition methods based on the use of optimized interface conditions on non-matching grids. The original Schwarz algorithm is based on a decomposition of the domain 17 into overlapping subdomains and the solving of Dirichlet boundary value problems in the subdomains. It has been proposed in [L 89] to use more general boundary conditions for the subproblems in order to use a non-overlapping decomposition of

232

Finite volumes for complex applications

the domain. The use of exact artificial boundary conditions as interface conditions leads to an optimal number of iterations, see [HTJ 88], [NRdeS 95]. As these boundary conditions are pseudo-differential, "low frequency" approximations of these conditions have then be proposed, see [D 93], [NR 95], [GGQ 96]. In [J 96] approximations which minimize the convergence rate of the algorithm are proposed, and increase dramatically the convergence speed of the method. The mortar method, first introduced in [BMP 89], which enables the use of non-conforming grids, can't be used easily with optimized interface conditions in the framework of Schwarz type methods. The goal of our work is to design and study a non-conforming domain decomposition method which allows for the use of Robin interface conditions (J^ +a) for Schwarz type methods. We consider the convection-diffusion equation

discretized by a finite volume method where rj and v are positive but arbitrary small and a is the vector field. We first consider the symmetric definite positive case (a = 0) and then in § 5 the convective case (a ^ 0). 2. Domain Decomposition at the continuous level

Let fi be a bounded domain in Rd for d > I and 77 > 0. We consider the following problem: Find u such that

The domain fi is decomposed into N non-overlapping subdomains, fi = ^Ji* 0, the above problem is reformulated as a domain decomposition problem: Find (ui)i
*

An iterative method for solving the above domain decomposition method is:

Numerical analysis

233

The well-posedness and convergence of the above problems and algorithm have been studied in [L 89]. We are interested in the discretization of (4) by a finite volume scheme with non matching grids on the subdomains's interface. 3. Finite volume discretization

The scheme is taken from [H 95]. On each domain 0; let Ti be a set of closed polygonal subsets of 0; such that f^ = \JxeTiK and £^ii the set of edges associated with Ti, i.e. a set of closed subsets of dimension d such that for any (K, K1} G T? with K ^ K1, one has either K n K' = 0, dim(K n K'} < d - 1 and K n K' G £ n .. In this case, dK n dK1 is denoted by [K,K']. We also assume that no edge intersects both dtli/dQ, and <9fi; n 80,. We shall use the following notations: Let 6i be an edge of £^ii located on the boundary of fii, K(ti) denotes the control cell K G Ti such that e; G K. £iD is the set of edges such that dtl n dtl{ = U ee £. D e. Let us recall that a Dirichlet boundary condition is imposed on this part of the boundary. £j is the set of edges such that d£li/d$l — Ueg^e. Let us recall that a Robin interface condition is imposed on this part of the boundary. £(K] denotes the set of the edges of K G Ti. £io(K] = £(K] n £iD is the set of the edges of K € Ti which are on <9Q n <9fV £i(K) = £(K}r\£l is the set of the edges of K 6 Ti which are on dtoi/dft. Afl(K) is the set of the control cells adjacent to K: Afi(K) = {K1 G Ti/ Kr\K' e £ n .}. We make the following Assumption 3.1 We assume that there exist points (y e ) ee £ n . on the edges (ye G €.) and points (xK)K&Ti inside the control cells such that for any adjacent control cells, K and K', the straight line [XK,XK>\ is perpendicular to the edge [K, K'] and [XK,XK'] H [K, K'} = {y[K,K']}> o,nd for any edge e G £i U £iD, the straight line [x/^( e ),y € ] is perpendicular to e. It is then possible to write a finite volume scheme for the equation (4). We shall use the primary unknowns (UK}K^T which aim at being approximations to U(XK}- The scheme is obtained by integrating (4) over each control volume K: This relation is discretized by

234

Finite volumes for complex applications

where meas([K, K1}} is the measure of [K, K'] and pt is a discretization (defined below) of the normal derivative du/driK on the edge [ K , K ' ] . For an edge [K, K'] common to two control volumes K and K1',

We have then the useful property that PK,K' = —PK',K- For an edge e on the boundary <90, the homogeneous Dirichlet boundary condition (5) is taken into account by

When there is no domain decomposition, this scheme has been analyzed in [H 95] in the more general case of discontinuous coefficients and is proved to be of order one for a discrete Hl-norm. 3.1 Discretization of the interface conditions On each interface edge e € E; of a control volume K — K(e) , we introduce pf and ut related by the relation

Then, the interface condition (6) on an interface edge e G £j of a control volume K — K(e) is discretized by

It will be useful in the error analysis to interpret (15) as a L2 projection. Indeed, let PQ(d£li/d£l] be the set of functions from dSli/d£l into E which are piecewise constant on the interface edges. To any discrete values (t' e ) ee £ { , we associate Ki((vf)e€£i) £ P°(dtti/d£l) its natural piecewise constant extrapolation:

The L2 projection on P°(dQi/d£l} is denoted by P{. With these notations, (15) is equivalent to

Numerical analysis

235

For simplicity, (18) will be (sometimes) denoted by abuse of notation

Theorem 3.2 The finite volume discretization defined by (11)-(12)-(13) -(14)(15) is well-posed. 4. Error analysis

For the error analysis, we need the following additional assumption on the interface edges (this assumption is relaxed in [AJNM 99]): Assumption 4.1 a) For any i and any e G £i, ye is the barycenter of e. b) For any i,j, <9Q;ndfij = ^ f £ £ { , t c d t i j t , i-e. d^liCid^tj can be written as the union of edges of £j and of Sj. Let ulK = U(XK) for any control cell K in T(Hj) and for any e € £j£>, let u\ = u(yf) = 0 and p\ — - j ^ ( y f ) . For any interface edge e £ £j, let p\ (resp. u\] be the mean value of -j^- (resp. u). With the notations of § 3.1, this can be rewritten as T i l ( ( p \ } t £ £ i ] = Pi(-j^) and ^((u*)^^) = Pl(u). Let FK be an approximation of order \K\O(h] of the integral of / on the cells K. The solution of the finite volume discretization with FK as a right hand-side is denoted by ulK for any control cell K in T(fJi) and p\ on any edge e G 8i U £io and u\ for any edge e 6 Si. We shall estimate the discrete errors elK — ulK — ulK, e\ — u\ — u\ and q\ = p\ — p\. As for Q, we take a = /3/h^ with 7 > 0. Theorem 4.2 We assume that the solution u of (2) is in C r 2 (A). Let us consider a family of admissible meshes T^ 0 s.t. Vz, V/c,Ve i ^ , y e 6 e,

236

Finite volumes for complex applications

5. Advection-diffusion problems We consider now the advection diffusion problem with a continuous velocity field a and a viscosity v :

Theorem 5.1 Under the same assumptions as in theorem 4-%> that a 6 C(ft) satisfies V • a = 0, we have the error estimate

ana

provided

6. Numerical results In order to illustrate the use of Robin interface conditions on non matching grids, 2-D test problems were performed with the finite volume scheme analyzed above. We solve the following problem:

with r] and v positive constants. The stopping criterion of the algorithm is that the max norm of the jump of the Robin interface condition is smaller than 10~8. The numerical solution is compared to an exact solution. We choose u(x,y) — x3 y2 + sin(xy). The right-hand side is obtained by applying the operator to u and using an approximate integration formula, the Gauss quadrature rule with four by four nodes per control volume. Thus the error due to inexact integration will be small with respect to the scheme error. A decomposition into 2x2 subdomains is considered. The L^ and discrete H^ (Theorem 5.1) norms are given for successively refined grids: • Case 1: TJ = 1, a = (0,0), v = 1. Initial grid: 9 x 9-8 x 8-7 x 7-6 x 6. • Case 2: r, = 1, a = (y, -x), v = le-2.

Initial grid: 9 x 9 - 8 x 8 - 7 x 7 - 6 x 6 .

Numerical analysis

237

For case 1 (diffusive case), the first order in the discrete H^ norm as expected from the theory seems to be attained only asymptotically for rather fine meshes. The error reduction between the finest two meshes is 1.88. The error reduction factor in the L^ norm is also improved with mesh refinement and seems better than O(h). On the contrary in case 2 (convective case), the error reduction factor deteriorates as the mesh is refined and seems to converge to the expected value \/2-

1/h

II

i ik

Hoc

H^ error reduction

8

0.0269608

0.0626146

16

0.0128424

0.0405976

1.54

32

0.0062293

0.0260017

1.56

64

0.00270477

0.0154682

1.68

128

0.00103779

0.00823524

1.88

Case 1

Table 1: Error vs. mesh refinement - No convection

1/h

II

Hoc

II

k

H\ error reduction

8

0.330713

0.142505

16

0.170031

0.0677237

2.1

32

0.0744696

0.0347217

1.95

64

0.0301526

0.0199572

1.74

128

0.0169002

0.0118619

1.68

Case 2

Table 2: Error vs. mesh refinement - Convection

7. Bibliography [AJNM 99]

ACHDOU Y., JAPHET C., NATAF F., MADAY Y., A new cement to glue non-conforming grids with Robin interface conditions, RI N° 419, CMAP Ecole Polytechnique, may 1999.

238

Finite volumes for complex applications

[BMP 89]

BERNARDI C., MADAY Y. AND PATERA A., A new nonconforming approach to domain decomposition: the mortar element method, Nonlinear Partial Differential Equations and their Applications, eds H. Brezis and J.L. Lions (Pitman, 1989).

[D 93]

DESPRES B. Domain decomposition method and the Helmholtz problem. II, Kleinman Ralph (eds) et al., Mathematical and numerical aspects of wave propagation. Proceedings of the 2nd international conference held in Newark, DE, USA, June 7-10, 1993. Philadelphia, PA: SIAM, 197-206 (1993).

[HTJ 88]

HAGSTROM T., TEWARSON R.P. AND JAZCILEVICH A., Numerical Experiments on a Domain Decomposition Algorithm for Nonlinear Elliptic Boundary Value Problems, Appl. Math. Lett., 1, No 3 (1988), 299-302.

[H 95]

HERBIN R., An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh, Numer. Methods Partial Differential Equations 11 , no. 2, 165-173, (1995).

[J 96]

JAPHET C., Optimized Krylov-Ventcell Method. Application to Convection-Diffusion Problems, DD9 Proceedings, John Wiley & Sons Ltd (1996).

[L 89]

LIONS P.L., On the Schwarz Alternating Method III: A Variant for Nonoverlapping Subdomains, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM (1989), 202-223.

[NR 95]

NATAF F. AND ROGIER F., Factorization of the ConvectionDiffusion Operator and the Schwarz Algorithm, M 3 AS, 5, n1, 67-93 (1995).

[NRdeS 95]

NATAF F., ROGIER F. AND DE STURLER E., Domain Decomposition Methods for Fluid Dynamics, Navier-Stokes Equations and Related Nonlinear Analysis, Edited by A. Sequeira, Plenum Press Corporation, pp367-376 (1995).

[GGQ 96]

GASTALDI F., GASTALDI L. AND QUARTERONI A., Adaptative Domain Decomposition Methods for Advection dominated Equations, East-West J. Numer. Math 4, 3, p 165-206 (1996).

Finite Volume Box Schemes

Jean-Pierre CROISILLE Departement de Mathematiques Universite de Metz F-57045 Metz Cedex 01 [email protected] univ-metz.fr

ABSTRACT: We present the numerical analysis on the Poisson problem of a mixed Petrov-Galerkin Finite Volume scheme for equations in divergence form div(/?(u, Vu) = f, which has been introduced in [CoC 98]. As the original box scheme of Keller, this scheme uses face centered degrees of freedom for the primal unknown u and for the flux (p. The underlying Finite Element spaces are the non-conforming space of Crouzeix-Raviart for the primal unknown and the div-conforming space of Raviart-Thomas for the flux. Optimal order error estimates are derived for the Poisson problem. Key Words: Finite- Volume method, Box Scheme, Box Method, Mixed Method.

1. Introduction The name of "box-scheme" is a generic denomination for several numerical schemes of different origins. It has been introduced primitively by H.B. Keller in the '70 on the 1-D heat equation, [Ke 71]. Generally speaking, the discrete equations are defined in a box-scheme from some kind of averages of the continuous equations on "boxes". Therefore, they are conservative schemes, i.e. schemes which guarantee, for equations in divergence form, an exact conservation of the flux at the level of the box. At least two variants of box-schemes are known in the litterature. The first one has been introduced in the '80 in compressible Computational Fluid Dynamics. As in Keller's scheme, the basic idea is that locating the degrees of freedom at the center of the faces instead at the center of the cells, could be more interesting for an accurate evaluation of

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conservative fluxes, [CDH 83], [CM 86], [Co 92]. Note that this kind of schemes has received much less attention in the CFD Finite Volume communauty than the cell-centered Finite Volume schemes. The second kind of box-schemes is known under the name of "box-method" or "finite volume element method". For the Laplace equation — Au = /, it consists of an approximation of u in a finite-element Pl or Q1-space. The discrete equations are defined by averaging the equation on a dual box surrounding each vertex of the mesh, [BR 87], [Ha 89], [CMM 91], [TS 93]. In this kind of scheme, two meshes are used. The primal one as support of the trial functions, and the dual one for designing the boxes for the discrete conservative equations. This design is in fact similar to the one of the cell-vertex Finite Volume method in CFD, [FS 89]. Concerning the numerical analysis of Finite Volume cell-centered methods, beside the exhaustive direct analysis of [EGH 97], there is by now an attempt in the FEM communauty for interpreting mixed Finite Elements methods as Finite Volume methods, [BMO 96], [Du 97], [YMAC 99]. The scheme presented here can be more or less attached to this kind of study. 2. The Finite Volume Box Scheme In [CoC 98], we introduced a new kind of box-schemes for equations in divergence form. As in box-schemes with finite-differencing interpretation, [Ke 71], [Co 92], the degrees of freedom are located at the center of the faces of the mesh. Nevertheless, the discretization is interpreted here as a Finite Element approximation, allowing to use Finite Element theory for the numerical analysis. Let us consider the 2D Poisson problem in mixed form with Dirichlet homogeneous boundary conditions

Suppose given a triangulation Th of the domain ft C R2 by triangles K. The number of triangles is NE. The number of internal edges, boundary edges are NAi, NAb and the total number of edges is NA = NAi + NAb. The Finite Element spaces that are used are u-space: The non-conforming Crouzeix-Raviart space with homegeneous boundary conditions Vh = P^c 0, equipped with the mesh dependent norm

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p-space: The div-conforming Raviart-Thomas space of least order Qh = RT°, equipped with the continuous norm

Recall that these spaces are Vh — {^/i/V K 6 Th, Vh\K £ Pl(K},Vh is continuous at the middle of each edge,^ = 0 at the middle of each edge on <9f7)

The scheme reads: find (w^p/J e Vh x Qh such that

In (2), the number of unknowns is IN A, since the global degrees of freedom for Uh, Ph are scalars located at the center of the faces of the mesh. The number of equations is clearly SNE + NAf,. A simple count of the faces (the edges) proves that in fact we have

Let us mention that coupling these two spaces is not standard in the mixed finite element methods, because they do not verify the Babuska-Brezzi condition, [Ba 71], [Bre 74]. 3. Numerical Analysis

3.1. Reformulation as a mixed Petrov-Galerkin method We consider the following mixed formulation of problem (1): find (u,p) £ HQ x #div such that for any (v,q) € L2 x (I/ 2 ) 2

or equivalently

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Applying the general Babuska theorem [Ba 71] onto mixed formulation we find easily that (5) is a well posed problem, whose solution is (u, Vw) , u G HQ r\H2 being the unique solution of the original problem (1). The scheme (2) appears now as a Petrov-Galerkin non conforming approximation of (5). Calling P° the space of constant functions in each triangle, it can be rewritten: find (uh,ph) & P^C]0 x RT° such that for any (vh,qh) e P° x (P°)2

or equivalently

Applying now the theory for mixed Petrov-Galerkin approximations, [Ni 82], [BCM 88], [BMO 96], [Cr 99], we get the following result: Theorem 1 (i) The scheme (7) does possess an unique solution (uh,ph) £ P^o x RT° verifying

(ii) We have the error estimate

3.2. Dual scheme Another mixed form of the Poisson problem linked with the bilinear form B is dual from (1): find (v, q) G L2 x (L2)2 such that for any (u,p) e HQ x H<&v

or equivalently

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Again by the Babuska theorem, problem (11) does possess an unique solution (v,q) = (u, Vw). The corresponding Petrov-Galerkin scheme is: find (vf^Qh) € P° x (P°)2 such that

As precedingly, we get the following result Theorem 2 (i) The dual scheme (12) does have an unique solution (t>/n

(ii) We have the error estimate

Note that (12) defines a non standard cell-centered Finite-Volume scheme for computing both the unknown and the gradient from the knowledge of the laplacian. 3.3. Second order error estimate Using the error estimates for the primal and dual schemes (7), (12), allows to derive a second order error estimate in the L2 norm for the unknown Uh in (7). The proof uses an Aubin-Nitsche like argument. Theorem 3 The solution (uh,Ph) °f Scheme (7) verifies the optimal error estimate

3.4. Further remarks A natural question is to ask whether there is the link between this scheme and the family of Finite Element mixed methods, [RT 77], [AB 85], [BDM 85], [BF 91]. In fact, it can be proved that the gradient part ph in (7) coincides with the mixed gradient p^ in [RT 77]. However, the Uh part in the mixed method

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is only a P° approximation of u. The optimal error estimate is therefore only of first order in the L2 norm. They are several methods in order to interpolate a posteriori Uh to an higher order approximation, [AB 85]. All these methods introduce a three variables problem (uh->ph, ^h), involving a new degree of freedom (a Lagrange parameter) A/,, at the interfaces of the mesh. It can be proved, [Cr 99], that one of these interpolations coincides in fact with the scheme (7). Moreover, contrary to the standard mixed method in its original formulation, the degrees of freedom for Uh and ph are decoupled, allowing to solve only an O(NA) system in Uh-, which is in addition symmetric definite positive. We refer to [CoC 98] for details onto the implementation of (7), and to [Cr 99] for the proofs of the results described here. 4. Conclusion Several works are devoted to the a posteriori interpretation of mixed Finite Element methods as Finite Volume ones for the primal unknown Uh- In other works, they are attempts to compute more easily this unknown, by introducing additional degrees of freedom. The advantage of the scheme (7) is that it is basically designed as a true Finite Volume scheme on a single computational cell, for Uh and ph. In addition, it gives a natural decoupling between the unknowns Uh and ph. Moreover, it has an optimal order of accuracy both for Uh and ph, without any post-processing or a posteriori interpretation. Note that this kind of schemes is not restricted to triangular meshes, or to the dimension 2. We think that the formulation of this scheme as a Petrov-Galerkin method, combining the advantages of mixed and Finite Volume methods, can be particularly interesting for computations involving complex fluxes. Moreover, it can be of some help for a better understanding of the link between mixed and cell-centered Finite Volume methods. Higher order extensions are currently in progress. Acknowledgments: The author acknowledges friendly B. Courbet, F. Dubois, and A. Debussche, J. Laminie, for stimulating discussions and encouragements.

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References

[AABM 98] B. ACHCHAB, A. AGOUZAL, J. BARANGER, J-F. MAITRE, Estimateur d'erreur a posteriori hierarchique. Application aux elements finis mixtes Numer. Math. 80, 1998, 159-179. [AB 85] D.N. ARNOLD, F. BREZZI, Mixed and non-conforming finite elements methods: implementation, postprocessing and error estimates, Math. Model, and Numer. Anal. 19,1, 1985, 7-32. [Ba 71] I. BABUSKA, Error-Bounds for Finite Elements Method, Numer.Math., 16, 322-333. [BR 87] R.E. BANK, D.J. ROSE, Some error estimates for the box method, SIAM J. Numer. Anal, 24,4, 1987, 777-787. [BMO 96] J. BARANGER, J.F. MAITRE, F. OUDIN, Connection between finite volume and mixed finite element methods, Math. Model, and Numer. Anal., 30,4, 1996, 445-465. [BCM 88] C. BERNARDI, C. CANUTO, Y. MADAY, Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem SIAM J. Numer. Anal., 25,6, 1988, 1237-1271. [Bre 74] F. BREZZI, On the existence, uniqueness and approximation of saddlepoint problems, arising from lagrangian multipliers R.A.I.R.O. 8, 1974, R-2, 129-151. [BF 91] F. BREZZI, M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer Series in Comp. Math., 15, Springer Verlag, New-York, 1991. [BDM 85] F. BREZZI, J. DOUGLAS, L.D. MARINI, Two families of Mixed Finite Element for second order elliptic problems, Numer. Math., 47, (1985), 217-235. [CMM 91] Z. CAI, J. MANDEL, S. McCoRMiCK, The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal., 28,2, 1991, 392-402. [CDH 83] F. CASIER, H. DECONNINCK, C. HIRSCH, A class of central bidiagonal schemes with implicit boundary conditions for the solution of Euler's equations, AIAA-83-0126, 1983. [CM 86] J.J. CHATTOT, S. MALET, A "box-scheme" for the Euler equations, Lecture Notes in Math., 1270, Springer-Verlag, 1986, 52-63.

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[Co 92] B. COURBET, Schemas boite en reseau triangulaire, Rapport technique 18/3446 EN, (1992), ONERA, unpublished. [Co 91] B. COURBET, Etude d'une famille de schemas boites a deux points et application a la dynamique des gaz monodimensionnelle, La Recherche Aerospatiale, n° 5, 1991, 31-44. [CoC 98] B. COURBET, J.P. CROISILLE, Finite Volume Box Schemes on triangular meshes, Math. Model. and Numer., 32,5, (1998), 631-649. [Cr 99] J-P. CROISILLE, Finite Volume Box Schemes and Mixed Methods, Preprint, Universite de Metz, 1999. [Du 97] F. DUBOIS, Finite volumes and mixed Petrov-Galerkin finite elements; the unidimensional problem. Preprint 295 du C.N.A.M., 1997. [EGH 97] R. EYMARD, T. GALLOUET, R. HERBIN, Finite Volume Methods, in Handbook of Numerical Analysis, vol. 5, Ciarlet-Lions eds., (1997). [FS 89] L. FEZOUI, B. STOUFFLET, A class of implicit upwind schemes for Euler equations on unstructured grids, Jour. of Comp. Phys., 84, 1989, 174-206. [Ha 89] W. HACKBUSCH, On first and second order box schemes, Computing, 41, 1989, 277-296. [Ke 71] H.B. KELLER, A new difference scheme for parabolic problems, Numerical solutions of partial differential equations, II, B. Hubbard ed., Academic Press, New-York, 1971, 327-350. [Ni 82] R.A. NICOLAIDES, Existence, uniqueness and approximation for generalized saddle point problems, SIAM J. Numer. Anal., 19,2, 1982, 349-357. [RT 77] P.A. RAVIART, J.M. THOMAS, A mixed finite element method for 2nd order elliptic problems, Lecture Notes in Math., 606, SpringerVerlag, 1977, 292-315. [TS 93] T. SCHMIDT, Box Schemes on quadrilateral meshes, Computing, 51 1993, 271-292. [YMAC 99] A. YOUNES, R. MOSE, P. ACKERER, G. CHAVENT, A new formulation or the Mixed Finite Element Method for solving elliptic and parabolic PDE, Jour. of Comp. Phys., 149, 1999, 148-167.

On nonlinear stability analysis for finite volume schemes, plane wave instability and carbuncle phenomena explanation

M. Abouziarov Research Institute of Mechanics Loba chevskey university, Nizhnii Novgorod, Russia e-mail [email protected]

1. Introduction

Courant restriction of integration time step, which follows from stability analysis of linearised Euler equations, does not provide the stability of some numerical methods for nonlinear problems . Some of these problems and attempt of analysis of the origin of these instabilities were made in [1]. In this report a procedure of the instability analysis for finite volume methods for the nonlinear equations and a method to correct the numerical schemes to avoid these instabilities are presented.

2. Nonlinear stability analysis

Assume for a 2D case a cell number (i,j) is integrated, where integer indexes i, j ( X and Y direction respectively ) are related with parameters (i.e. grid, pressure, density, velocities, energy) at the cell centers and half-

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pressure and velocity perturbations are small, but there can be the density discontinuity. For the calculation process to be stable the perturbation of the new velocity is to be equal to the old one or this velocity perturbation is to be diminished step by step.

Assume:

respectively

and

and etc.

(values with covers are perturbations of the appropriate values). For the original Godunov?s method, using linearised Riemann?s solution with the different left and right densities and sonic velocities, the following simple estimation can be obtained:

A A similar formula is for unew For the second order modification [3], the formulas

are analogous. For linearised Euler equations, when

, it is

obvious from (6) that the Courant restriction is enough for stability. But when there is the density discontinuity and the change of the density in the cell during one step is big, to provide the calculation stability the time step is to be stronger restricted or the scheme is to be reconstructed to correct these additional instability terms; without it even small perturbations of the velocity may to increase. It can be shown that on the back of the shock wave instability estimations are similar (6) but with stronger possible instability. For example, such instability situations are possible on the back of a strong shock wave in a wind tunnel (plane wave instability) or near the head of the shock wave, generated by the sphere or other body in the supersonic flow ?carbuncle phenomena?. In the ?carbuncle phenomena?, at the beginning of the iterations, the situation is very similar to the plane wave instability case - a strong

Numerical analysis integer indexes show the parameters on the cell boundaries.

Eq.l represents the mass conservation law for this cell. For simplification (without any restrictions) the grid is assumed to be stationary and regular. °' '

are

respectively volume , surface area of a boundary of this cell and integration time step; u and v are velocities for X and Y directions respectively; and the values with the upper indexes indicate the new time step level.

Eq.2 represents the momentum conservation law in Y direction for this cell (similar for X direction). Using the expression for

ot

^J from eq. 1, it follows from Eq.2 ,

Eq.4 is a formula for obtaining the velocity on the new time step based on the velocity for the previous one. The formulas for velocity in the X direction and for pressure derived from the energy equation have a similar form. These formulas are nonlinear and follow from the conservation laws. Using different ways for obtaining the fluxes on the cell boundaries, different numerical schemes can be constructed. Let us consider the behavior of (4) for the plane wave in the X direction in the vicinity of the density discontinuity for widely used Godunov?s method. Assume that our flow is near to equilibrium in its pressure and velocity and the values of

249

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shock wave is separated from the body and goes ahead.

3. A possible way of correction

A possible way to correct this instability follows from formula (6): it is necessary to eliminate the perturbations which appear in addition to the standard Courant perturbations. These corrections should have the following logic: corrections are to have higher order accuracy than that of the numerical scheme, they are not to change the approximation error; they are to correct only the instability terms to avoid the increase of the perturbations. One way to make the scheme stable is to use appropriate corrections of instability terms after integration, but in this case it is necessary to analyze the other kind of discontinuity problem more accurate (shock wave) too, which is rather difficult. The second way to eliminate instability terms is to correct such terms in the fluxes [2]. For the momentum equations (3) the correction term is as follows:

where the values with the covers (respective errors) are to have the second order accuracy in the appropriate directions. For example:

Numerical analysis

The momentum fluxes for the X direction and energy fluxes are corrected in a similar way. With these corrections, the original Godunov method and its second order accuracy modification [3] become stable with standard Courant stability restrictions for those schemes and do not have the plane wave instability and ?carbuncle phenomena? troubles. In (5) it was assumed that near the pressure and velocity equilibrium state the absolute values of the appropriate perturbations are the same for different cells; it is reasonable because the relative truncation error for all values is the same and depends only on the type of a computer. From our density discontinuity analysis it follows that the velocity and pressure perturbations do not compensate each other after integration in formula (4) because the flux perturbations are proportional to the appropriate densities, which are different. For a shock wave; the situation is more complicated - absolute perturbation values are different and are related with the absolute values of the respective fields.

4. Conclusions

Thus, it may be concluded that: in constructing numerical methods and analyzing their stability for computer calculations in the regions of possible big gradients, it is necessary to take into account that only relative truncation error is the same for all numbers in the computer, whereas absolute values of perturbations are proportional to this relative truncation error and to those absolute values.

5. Bibliography

[1] J. QUIRK, "A contribution to the great Riemann solver debate," International Journal for numerical methods in fluids 18 (1994), pp. 555-574. [2] ABOUZIAROV M. Nonlinear stability analysis of Euler equations for predictor corrector schemes. // Godunov?s method for gas dynamics: Current Applications and Future

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Developments, A Symposium Honoring S.K. Godunov, May 1-2, 1997, The University of Michigan, Ann Arbor, Michigan [3] ABOUZIAROV M. On accuracy increasing of Godunov's method for nonlinear problems of continuum mechanics. // Godunov?s method for gas dynamics: Current Applications and Future Developments, A Symposium Honoring S.K. Godunov, May 1-2, 1997, The University of Michigan, Ann Arbor, Michigan

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A comparison between upwind and multidimensional upwind schemes for unsteady flow

P. Brufau and P. Garcia-Navarro Centra Politecnico Superior 50015 Zaragoza, Spain

ABSTRACT An ideal scheme for the solution of multidimensional non-linear systems of partial differential equations governing fluid motion has not yet be found despite years of research effort and many scientific contributions. In the context of finite volume approximations based on flow dependent schemes, discussion is opened as to whether upwind or multidimensional upwind schemes are preferable for solving 2D problems. In this work we consider the use of upwind and multidimensional upwinding techniques for 2D shallow water flows, in particular, to the simulation of dam break flows. The basis of the two numerical methods is stated and the particular adaptation to the shallow water system is described. As test cases, laboratory experimental data supplied by partners of the Working Group on Dam Break Flow Modelling (CADAM) are used. Key Words: upwind schemes, shallow water, dam break.

1. Introduction Upwind schemes[l, 9] are accepted as accurate methods for the numerical solution of systems of conservation laws in one dimension. Attempts of applying these techniques in higher dimensions have generally relied on essentially one-dimensional algorithms combined with some form of operator splitting. Indeed, the standard extension of the finite volume methods is still to solve onedimensional Riemann problems created at the interfaces of the finite volumes by the discontinuities in the reconstruction stage of the algorithm. When extended in this way many features of the flow, particularly if they are not aligned with the grid, can be completely misinterpreted by the numerical scheme. Discussion is open as to whether the schemes based on ID Riemann solvers

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are the most suitable choice for multidimensional calculations because they seem inadequate for capturing 2D flow features. Recently, a class of upwind methods has emerged which attempts to model the equations in a genuinely multidimensional manner[2, 3, 5, 6]. These schemes are designed to monitor the average time evolution of the approximation to the solution within a complete grid cell rather than concentrating on the activity at the interfaces. Multidimensional upwind schemes were developed initially for the approximation of steady state solutions of the two-dimensional Euler equations on unstructured triangular grids, although they could be applicable to any system of hyperbolic conservation laws. One such case is given by the shallow water equations[4]. In this paper the performance of an upwind and a multidimensional upwind technique for 2D shallow water flows is described in first order accuracy. In the next sections, the basis of the numerical methods is stated and the application to the simulation of 2D dam break flows is presented. 2. Governing equations

Depth averaging of the free surface flow equations under the shallow water hypothesis leads to a common version of the 2D shallow water equations which, in conservative form is,

with

where U represents the vector of conserved variables (h depth of water, hu and hv unit discharges along the coordinate directions), F and G are the fluxes of the conserved variables across the edges of a control volume. They consist of the convective fluxes together with the hydrostatic pressure gradients. H is the source term. In addition, u and v are the velocities in the x, y directions respectively, g is the acceleration due to the gravity, SQX, SQy are the bed slopes and Sfx, Sfy the friction terms in the x, y directions. For the friction term, the Manning equation has been used.

3. Numerical models

3.1 Upwind method A cell centered finite volume method is formulated for equation (1) over a triangular control volume where the dependent variables of the system are represented as piecewise constants. The integral form of (1) for a fixed area 5 is

and, applying the divergence theorem to the second integral, we obtain

where C is the boundary of the area 5, and n is the outward unitary normal vector. Given a computational mesh defined by the cells (volumes) of area Si, where i is the index associated with the centroid of the cell in which the cellwise constant values of U are stored, equation (4) can be represented by

where a mesh fixed in time is assumed. The contour integral is approached via a mid-point rule, i.e., a numerical flux is defined at the mid-point of each edge, giving

where Wk represents the index of edge k of the cell, NE is the total number of edges in the cell. The vector nwk is the unit outward normal, dCWk is the length of the side, and (F, G)^ fc is the numerical flux tensor. The evaluation of the numerical flux in equation (6) is based on the Riemann problem defined by the conditions on the left and right sides of the cell edges. An important feature of the 1D upwind schemes for non-linear systems of equations is exploited here. This is the definition of the approximated flux jacobian, Ai+i, constructed at the edges of the cells and satisfying special conditions[7]. The 2D numerical upwind flux in equation (6) is obtained by applying the expression of the numerical flux across the interface i + \ of a cell in a 1D domain to each edge Wk of the computational cell in a 1D form. The ID philosophy is followed along the normal direction to the cell walls, making use of the normal numerical fluxes, so that

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R and L denote right and left states respectively at the Wk edge, (ARL) represents the approximate Jacobian of the normal flux. As suggested by Roe[7] the matrix ARL has the same shape as A, being

but is evaluated at an average state[l]. We can now substitute (7) into (6), so that (5) can be written as

which is an ordinary differential equation and can be integrated by standard methods such as a forward Euler time integration procedure.

The stability criterion adopted has followed the usual in explicit finite volumes [1]

where dij is the distance between the centroid of the cell i and its neighbours. 3.2. Multidimensional upwind method The application of this technique to the shallow water system requires the equations to be written in terms of the conserved variables, so that (1) can be expressed in its non-conservative form as

in which only the homogeneous part has been considered. It is useful to express the equations in terms of the primitive variables in a non-conservative way, as

In the conservative formulation, the fluctuation is defined as

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259

We can use the relation between the two sets of variables to define new matrices R and S,

so that Provided that the variables V are linear over the cells T, the gradients, Va and Vy, are constant, and we can write the fluctuation as with the definitions:

We approximate R, S by where the averaged variables are simply

summing over the nodal values at the vertices of the triangle T. Note that with this definition of R, S we are only approximating equation (18). As a consequence, we lose conservation in the numerical evaluation of the fluctuation. The next step consists on computing the fluctuations and distributing them to the vertices of every cell by means of an advection scheme[3]. This last step requires the description of wave models[8] and the choice of an advection scheme. 4. Numerical results Results obtained with first order upwind and multidimensional upwind approximations on unstructured Delaunay triangular meshes for the experimental test case proposed by Prof. Zech (Civil Engineering Dpt., UCL Belgium) from the Working Group on Dam Break Flow Modelling are going to be presented. The test combines a square shaped upstream reservoir and a 45° bend channel (see Fig. 1). The flow will be essentially two-dimensional in the reservoir and at the angle between the two straight reaches of the 45° bend channel. Two features of the dam break resulting flow are of special interest: the damping effect of the corner, and the upstream moving hydraulic jump which is formed by reflection at the corner.

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The initial conditions are water at rest with the free surface 25cm in the upstream reservoir and 1cm water depth in the channel. All boundaries are solid non-slip walls except the outlet which is considered free. The Manning coefficient is n^ = 0.0095 for the bed. The number of elements used in the mesh is 15397. Nine gauging points were used in the laboratory to measure water level in time. Their location is shown in Fig. 1. The measurements at these stations are compared to the numerical results and displayed in Fig. 2. In Fig. 3 the free surface is described at time 10s and finally, Fig. 4 represents snapshots of the free surface at time 18s. In general, the figures indicate a good performance of the two numerical schemes. The arrival times of the main shock fronts is better captured by the upwind method. Some differences are noticeable in P3 and P4 as the reflected shock front celerity is concerned. This may be attributed to the treatment of the boundary conditions. The great difference between the results obtained with both schemes is referred to the free surface plot. Perhaps measurements along the walls of the channel should be taken into account to demonstrate which approximation is better. Till now, only data in the central axis have been measured. 5. Conclusions

An upwind and a multidimensional upwind scheme for the solution of the 2D shallow water equations has been applied in first order accuracy for dam break modelling. The numerical results have been validated by comparison with experimental data in one test case. Differences on the results obtained with both techniques do not follow a clear tendency and it is difficult to establish the superiority of one over another. In the test case presented, both results fail to reproduce exactly the arrival times of the reflected wave (P2 and P3 gauge points). This suggests that the reflection at the corner may require an improved numerical treatment. Moreover, differences can be noticed at the free surface plots. Reflected waves from the walls are clearly observed in the multidimensional upwind results and perhaps a good test could be performed taking measures along the channel walls to give a new point of view in the numerical analysis.

6. Bibliography

[1]

[2]

BRUFAU P., An upwind scheme for the 2D shallow water equations with applications , Num. Anal. Rep. 11/97, University of Reading, England (1997).

BRUFAU P. AND GARCIA-NAVARRO P., Two dimensional dam break flows in unstructured grids, Hydroinformatics98,

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Copenhague, Denmark (1998). [3]

DECONINCK H. et al., Multidimensional Upwind Methods for Unstructured Grids, Unst. Grid Met. for Advec. Domin. Flows, AGARD, 787(1992).

[4]

GARCIA-NAVARRO P. et al., Genuinely Multidimensional Upwinding for the 2D Shallow Water Equations, J. Comp. Phys., 121 (1995), p. 79-93.

[5]

HUBBARD M.E., Multidimensional Upwinding and Grid Adaptation for Conservation Laws, PhD Thesis, University of Reading, England, 1996.

[6]

PAILLERE H., Multidimensional Upwind Residual Distribution Schemes for the Euler and Navier-Stokes Equations on Unstructured Grids, PhD Thesis, University of Brussels, Belgium, 1995.

[7]

ROE P.L., A Basis for Upwind Differencing of the TwoDimensional Unsteady Euler Equations, Num. Met. Fluid Dyn. II (1986).

[8]

RUDGYARD M., Multidimensional Wave Decompositions for the Euler Equations, VKI Lecture Series, Comput. Fluid Dynam. (1993).

[9]

SLEIGH P.A. et al., An Unstructured Finite Volume Algorithm for predicting flow in rivers and estuaries, Comp. and Fluids (1997).

7. Figures

Figure 1: Plane view of the channel.

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Figure 2: Water depth history at points a) P3, b) P4, c) P5 and d) P9.

Figure 3: Free surface at time t=10s with upwind (a) and multidimensional upwind schemes (b).

Reformulation of the unstructured staggered mesh method as a classic finite volume method Blair Perot & Xing Zhang University of Massachusetts 219 Engineering Laboratory Amherst, MA 01003

ABSTRACT A generalization of the Harlow & Welch (1965) staggered mesh method to twodimensional unstructured meshes is presented. With certain choices of the interpolation operators, it is shown that this method can be recast as a classic finite volume method using a single set of non-overlapping control volumes and collocated variables. When the divergence form of the Navier-Stokes equations are discretized using the unstructured staggered mesh method the resulting equations are equivalent to a classic finite volume method for the velocity vector. When the rotational form of the Navier-Stokes equations are discretized using the unstructured staggered mesh method the resulting equations are equivalent to a classic finite volume method for the vorticity vector. Key Words: unstructured, staggered mesh, reformulation, Navier-Stokes equations.

1. Introduction The Cartesian staggered mesh method has a number of mathematical properties that make it a popular choice for simulations of incompressible fluids. In particular, the method does not have spurious 'pressure modes' and does not require stabilization or damping terms to control unphysical small-scale pressure fluctuations. In addition, the method is known to conserve mass, momentum, total energy, kinetic energy and vorticity. The latter two conservation properties are not found in generic control volume approaches and are particularly important in direct and large eddy simulations of turbulence where the cascade of turbulent kinetic energy (or enstrophy) from large to small scales (or vice versa) is critical to the overall predictions of the turbulence behavior.

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The success of the Cartesian staggered mesh method originally developed by Harlow & Welch [HAR 65] has motivated the search for generalizations of the method to unstructured meshes. While such a generalization is a non-trivial exercise, the unstructured staggered mesh methods of Porsching [AMI 81], and Nicolaides [NIC 93] have demonstrated many of the attractive properties of the Cartesian staggered mesh method by taking advantage of the fact that every unstructured mesh has a locally orthogonal dual mesh - the Voronoi tesselation. Chou [CHO 97] has shown the connection of the unstructured staggered mesh methods to nonconforming finite element methods. In this paper we discuss the direct connection with classic finite volume methods. It is via this connection with classic finite volume methods that the hitherto uninvestigated conservation properties of unstructured staggered mesh methods can be evaluated.

Figure 1. Two dimensional unstructured mesh and the dual Voronoi mesh.

2. Analysis of the Divergence Form The unstructured staggered mesh discretization is simply a way of forming discrete difference operators. It is actually independent of the equations to which it is applied. Hence, different discretizations of the Navier-Stokes equations are possible depending on which form of the equations are discretized. In this section, we will look at unstructured staggered mesh discretizations of the divergence form of the Navier-Stokes equations.

Discretizations based on the divergence form of the equations are of interest because they are able to discretely conserve momentum. While momentum conservation is a trivial consequence of a classical finite volume method, it is not an obvious trait of staggered mesh methods. This is because the staggered mesh methods only updates the normal velocity components at cell faces, tangential

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velocity components are interpolated not evolved. It will be shown that with certain choices of the interpolation operators, the staggered mesh update of face normal velocities is directly equivalent to a classic finite volume method which updates the velocity vector at cell centers. 2.1 Discretization of the Divergence Form The normal vector at each face is assumed to point from cell Cl to cell C2. At boundary faces the normal vector is assumed to point out of the domain and cell C2 is a virtual cell located at the domain boundary. The discrete equation for the evolution of the normal velocity component is then given by,

where c t = - ^ - £ u f u A f

is a conservative discretization of the convection term cell faces

evaluated

in each

cell,

dc =^- !>(Vu + Vu ) - n f Af

is a conservative

discretization of the diffusion term evaluated in each cell, V. is the volume of each cell, Af is the face area, Wf is the distance between neighboring cell circumcenters, and W^ is the distance between the face circumcenter and the cell circumcenter. Note that u = u • nf is the normal velocity component at each cell face and u is the normal velocity component that points out of a particular cell. Similarly nf is the normal vector pointing out of a particular cell. 2.2 Reformulation as a Classic Control Volume Scheme The reformulation of the divergence form is accomplished by multiplying each evolution equation for the face normal velocity component (Eqn. [2]), by the face normal vector, and summing over all the faces in the computational domain. This results in the following equation,

The goal is then to recast this into a form that looks like a summation over control volume cells. Recognizing that Wf = W^ + W^ and that W^ = 0 at

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boundary faces, and also noting that at boundary faces pc2 = pf, rewritten as a summation over cells.

[3] can be

This can be further simplified using three identities derived from Gauss' divergence theorem. Gauss' Divergence Theorem for an arbitrary bounded volume and a vector quantity f is,

where Q is the volume and dQ, is the boundary of the volume with unit normal vector n oriented outwards from the volume. We are actually interested in convex polygonal volumes where Gauss' Theorem simplifies to,

cell faces

If f is a nonzero constant vector then £ nf Af = 0 . If f = (x • a)b where a and b are nonzero constant vectors and x is the position vector with an origin located at the cell circumcenter then it follows from [6] that in two-dimensions cell faces

X n r n f W c f A = VCI where I is the identity matrix. Finally, if f = (a-x)u where u

is the velocity vector and a is an arbitrary nonzero constant vector, then Gauss' Theorem gives,

The gradient of the position vector is the identity matrix ( x s . = 5sj), and since a is an arbitrary vector

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where u is the outwards normal component of the velocity at the cell faces. This is an exact equation for polygonal volumes. If we assume that the velocity field u is a constant function (a first order approximation), then the second term will be zero and the integrals can be evaluated. In two dimensions, this gives the relation,

The first two identities are geometric and they are exact. The last expression (Eqn. [9]) is really not an identity, it is a first order approximation for the cell velocity vector given the normal velocity components at the cell faces. With these geometric identities and this definition for the cell velocity vector, [4] becomes,

This equation is true for a collection of cells, but it is also true for a single mesh cell. The preceding analysis makes no distinction as to the number of cells. Applying the previous definitions for cc and dc we can therefore write that

This is true for each mesh cell, and has the form of a classic control volume scheme for the velocity vector in the mesh cells. It is important to note however, that despite the apparent similarity there remains a subtle distinction from classic control volume schemes. In the staggered mesh scheme, the normal velocity component u is the primary unknown and uc is a derived quantity. In classic control volume schemes, uc is the primary velocity unknown and the normal velocity component at faces is derived. 3. Analysis of the Rotational Form In this section, we will look at unstructured staggered mesh discretizations of the rotational form of the Navier-Stokes equations.

where u is the velocity vector, co is the vorticity, pd = p + y u - u is the specific dynamic pressure, and v is the kinematic viscosity. This equation assumes that viscosity is constant, but it is otherwise equivalent to other forms of the incompressible Navier-Stokes equations. Variable viscosity diffusion can be still be

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represented in rotational form but the extra term (involving second derivatives of viscosity) complicates the analysis unnecessarily. This particular form of the Navier-Stokes equations is of interest because it appears to be inherently suited to the staggered mesh discretization. The classic staggered mesh method can be rearranged to look like a discretization of [12]. It will be shown that in two dimensions the staggered mesh update of face normal velocities is directly Figure 2. Notation for a cell face in relation equivalent to a classic finite to neighboring cells and nodes. volume method which updates the vorticity at nodes and where the control volumes are the dual mesh Veronoi polyhedra. 3.1 Discretization of the Rotational Form Using the rotational form of the Navier-Stokes equations, the normal component of the face velocity is discretized as,

where con is the vorticity at a node in the direction out of the two-dimensional plane. The face tangential points from node Nl to node N2 and is oriented 90 degrees counterclockwise to the face normal vector. The tangential velocity at the nodes in the convection term is given by v n = u n • t f . 3.2 Reformulation as a Control Volume for Vorticity The reformulation of the rotational form is accomplished by dividing each normal velocity evolution equation (Eqn. [13]) by the face area and then multiplying by -1 if the face normal points clockwise with respect to the node in question, and finally summing over all the faces touching a specific node. The result will be shown to be a control volume equation for the vorticity. In mathematical notation we start with the following equation,

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where the normal vector at each face has been chosen to point in a direction counterclockwise with respect to the node in question. In addition, node n0 is the node around which the summation is occurring and node ni is the other node adjoining that face. In this case, we use a discrete version of Stokes Curl Theorem to simplify the equations. Stokes theorem says that for an arbitrary bounded surface and vector quantity f,

where S is the surface with normal z, 3S is the boundary- of the surface, and the integration takes place in a counterclockwise direction around the boundary with respect to the face normal. We are actually interested in the planar polygonal Veronio regions surrounding each node, it which case Stokes Theorem simplifies to

If we let f equal the velocity vector and make the first order assumption that the velocity is constant in the Veronio cell then we obtain,

where An is the area of the Veronio cell surrounding the node, and the face normal vectors are assumed to point in a counterclockwise direction around the node. In conjunction with [17] it is clear that for interior nodes, the pressure term is identically zero. The net result is that [14] can be written as,

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where the face vorticity flux is given by (vco) lf = T(co n0 u n0 + co nj u ni ) • tf and tf points out of the Veronio cell. This is a discrete version of the continuous two-dimensional vorticity evolution equation,

Again, it is important to note that despite the apparent similarity there remains a subtle distinction from classic control volume schemes. In the staggered mesh scheme, the normal velocity component u is the primary unknown and con is the derived quantity. In a classic control volume schemes, con would be the primary unknown. So the staggered mesh scheme differs from a standard vorticitystreamfunction or vorticity-velocity formulation in the fact that (often complex) boundary conditions on the vorticity are not required. 4. Discussion The primary result of the current work is that staggered mesh methods are not just control volume methods applied on staggered control volumes, but are directly equivalent to classic collocated control volume methods. It was shown that unstructured staggered mesh discretizations of the divergence form of the NavierStokes equations are equivalent to classic control volume method for the velocity vector in mesh cells. Likewise, unstructured staggered mesh discretizations of the rotational form of the Navier-Stokes equations are equivalent to classic control volume method for the vorticity vector at mesh nodes (in Veronio cells). These equivalencies imply that the method possesses certain conservation properties. References [HAR 65]

HARLOW, F. H. & WELCH, J. E., Numerical calculations of time dependent viscous incompressible flow of fluid with a free surface, Phys. Fluids, 8 , 1965, p. 2182.

[NIC 93]

NICOLAIDES, R. A., The covolume approach to computing incompressible flow, Incompressible Computational Fluid Dynamics, M. D. Gunzburger & R. A. Nicolaides, eds., Cambridge University Press, 1993, p. 295-234.

[AMI 81]

AMIT, R., HALL, C. A. & PORSHING, T. A., An Application of Network Theory to the Solution of Implicit Navier-Stokes Difference Equations, J. Comput. Phys. 40, 1981, p. 183.

[CHO 97]

CHOU, S. H., Analysis and convergence of a covolume method for generalized Stokes problem, Math, of Comput. 66 (217), 1997, p. 85.

A mixed FE - FV algorithm in non-linear solid dynamics

Serguei' V. Potapov EDF, Research and Development Division Acoustics & Vibration Mechanics Branch 1, avenue du General-de-Gaulle, B.P. 408 92141 Clamart Cedex France

ABSTRACT. This article presents an extension to the transient non-linear solid dynamics of a mixed Finite Element-Finite Volume algorithm initially proposed for simulation of hydrodynamic problems with compressible fluids. This algorithm is based on an Arbitrary Lagrangian Eulerian (ALE) formulation of motion. By using a fractional step method, the ALE problem is divided in two phases : a Lagrangian one and a transport one. The Lagrangian phase dealing with spatially symmetric terms is approximated using a conventional Galerkin Finite Element formulation. To deal with the transport phase, whose operator is not symmetric, the Finite Volume formulation is applied. Using two kinds of finite volume cells warrants a compatibility of spatial approximations of the Lagrangian and transport phases and allows to perform correctly the transport of all unknown variables. Key Words : non-linear solid dynamics, finite element, finite volume, explicit scheme

1. Introduction The ALE formulation is well established in fluid dynamics where it is applied to the transient analysis of flows on moving meshes. The capacity of the ALE kinematics to manage strong fluid flows on dynamically deforming meshes is attractive to use in the non-linear solid mechanics dealing with large elastic-plastic deformations. However, this extension is not trivial because the stress tensor of solids is not only function of instantaneous quantities (as for fluids) but depends on the specific history of each material point. In order to evaluate the stress state on a current step, the constitutive equation must be integrated in time. Since in the ALE formulation

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the computational grid and the material move independently, the quadrature points at which the stresses are evaluated coincide throughout the deformation process with different material points, having in general different deformation histories. Thus, the stress transport procedure is needed to overcome the numerical difficulties due to relative motion between grid nodes and material particles. Furthermore, the same update procedure must be applied to each internal variable of the elasto-plastic analysis, e.g. to the yield stress in isotropic hardening or to the back stress in kinematic hardening. Direct time integration of the ALE equations is not easy because they contain the spatial derivative terms corresponding to very different physical phenomena. So, we use a fractional step method to integrate in time such equations. The ALE problem of each time step is split into two phases. The first one is so called Lagrangian phase, in which the computational grid is assumed to follow the material points, what cancels all transport terms. The second phase is a convective one in which only the transport terms are taken into account. The resolution of ALE problems on unstructured grids is generally made via the weak formulation suited by the finite element approximation. However, the standard Galerkin finite element approach produces spurious numerical oscillations when applied to non-symmetric operator of the transport phase. Therefore, the upwind technique must be used to overcome these instabilities. In [HUE 95], two distinct strategies, borrowed from the fluid dynamics experience, are presented. A Lax-Wendroff scheme and a Godunov algorithm are implemented in finite element context to perform the transport of stresses and stress-related variables. To stabilise the momentum equation some ad-hoc donor-element upwind technique is used. Here, we use the finite volume technique to perform, in unified and conservative way, the convection of all unknown variables [POT 97]. 2. A mixed approach using Finite Element and Finite Volume formulations First, we rewrite the standard ALE equations under a conservative way reinforcing the momentum, the energy and the solid time dependent constitutive equations by the mass conservation equation. Then, the ALE system is split into two sub-systems. All symmetric terms like local strains of the continuum and diffusion terms are grouped in the first sub-system which corresponds to the Lagrangian phase. The second subsystem contains only purely convective terms describing the motion of the material through the ALE computational grid. It corresponds to the transport phase. Such a splitting is very general and allows to consider both geometric and material nonlinearities related to the problem. Furthermore, it enables us to choose for each phase the appropriate integral formulation and method of resolution. 2.1 Finite Element formulation of the Lagrangian phase The equations of the Lagrangian phase possess a self-adjoint (symmetric) spatial operator. To guarantee the best approximation, we apply to thein a conventional Galerkin integral formulation with symmetric weigh functions Wp,Wv,We. Due to the fact that all equations of the ALE formulation contain convective terms, we must

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consider the mass density p, the material velocities t>,, the internal energy etnt and Cauchy stresses

• momentum equation

• internal energy equation

• constitutive law

In the constitutive relations, Cijki is the material response tensor which relates any frame-invariant rate of the Cauchy stress to the velocity strain tensor DIJ ; the tensor Sijki acting on the spin tensor uiki assures the objectivity of the Cauchy stress tensor. It should be noted that pressure gradients describing the propagation phenomenon (acoustic waves) and evaluated usually in fluid dynamics algorithms by approximate Riemann solver together with pure convection fluxes are included here to the Lagrangian phase. The reason for it is that in non-linear solid materials, the propagation phenomenon is more complex then in Newtonian fluids because of coupling of constitutive equations. Thus, it is problematic to construct a very general algorithm based on the approximate Riemann solver capable to deal with any non-linear solid material. A finite element formalism is then used. The computational domain Q C JR3 is approximated by a discretisation Ph using tetrahedrons or brick elements Pe £ Ph • The following discrete spaces are also introduced :

where PI is the space of polynomials in three variables and of degree 1 ; Vh is the approximated value of the velocity on the discretization Ph '•> u/, is the m-dimensional vector containing approximate values of p, eint, stresses and internal variables.

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2.2 Finite Volume formulation for the transport phase

After splitting, the sub-system of the transport phase can be written in the following conservative vector form :

where V 6 Rmx]Q,tfinal] -> ]Rm,F(V} e JRm ->> Rm. The spatial operator of the transport phase sub-system is non-symmetrical. Therefore, an upwind-like procedure must be applied . Dealing with unstructured grids, we use the finite volume method because of its ability to solve hyperbolic problems in a conservative form using flux computations at the cells boundaries. The direct application of a finite volume technique with only one kind of cells (like cell or node centred cells) is not optimal for our ALE algorithm. On one hand, the difficulty lies in the way the physical variables are represented. In the Lagrangian phase some of them are represented with a constant approximation per element like stresses whereas others have a linear approximation such as velocities and displacements. Therefore, a complete use of a cell-centred technique in the transport phase requires a double averaging operation to pass from the nodal representation to the constant per element one and vice-versa, which gives too much diffusion. On the other hand, the necessity to govern the motion of the deformable computational grid does not enable us to use the same spatial location for velocities and element-centred variables, like used in conventional Computational Fluid Dynamics (CFD) on fixed grids. It is due to the fact that the ALE grid rezoning techniques usually deal with nodes attached quantities (for instance, when solving the Laplace equation to regularise the ALE grid). In order to avoid such difficulties, we use two kinds of cells (cell and node centred cells) in a combined finite volume algorithm of the transport phase. The momentum is localised on node-centred cells built around finite element nodes of the discretisation Ph, whereas the other conservative quantities are integrated using cells coinciding with the finite element mesh Pe. To approximate the momentum, the third discrete space is introduced:

where C/ are node centred cells forming an other covering of the domain f2. A variational formulation of the transport phase can be written in general form as follows : Find Vh that satisfies :

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Choosing W'/, in an appropriate way in the spaces Uh or W/j and using Green's formula, we can rewrite (2) on a local level as follows : • momentum equation

with Vi = ( pvi ) , Fik — ( pvi(vk-vk)

), on the node-centred covering and

• other quantities

with

integrated using cell-centred cells coinciding with the finite element mesh Pe. Such integral forms present an advantage to be compatible with the spatial approximation adopted in the Lagrangian phase. Moreover, they allow to deal with moving and deformable grids. To evaluate the convective flux integrals at the interfaces between neighbour polygonal cells, we suit the standard procedures described by many authors (see for example [NKO 94]). However, due to the original split technique presented above, the construction of numerical convective fluxes is simpler than in standard CFD calculations. Indeed, the solution of the Riemann problem is quite simple because the transport phase system is an hyperbolic degenerated system. Thus the Roe scheme based on characteristic variables cannot be applied. Since the upwind direction is uniquely determined by the quantity (v-v).n denoting the scalar product between a convective velocity across the cell boundary and an outward normal to this boundary, we can use the well known full upwind scheme which respects the flow direction. 3. Numerical examples The efficiency of the mixed convection technique presented above has been evaluated on two solid dynamics problems dealing with elasto-plastic material. 3.1 Stress wave propagation problem

The first example is a ID elastic wave propagation problem first considered in [LIU 86]. A stress wave is generated in a long rod by a pressure pulse applied at one end starting att = 0. The rod is discretised in 400 constant-width elements over a length sufficient to avoid wave interactions with the unloaded end of the grid during the whole simulation. The ALE grid is kept fixed until t = 24 sec, then suddenly moved with a positive constant velocity until the final time t — 32 sec. The definition of the problem is following :

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To provide a severe test of the proposed transport algorithm, the grid velocity is taken equal to as much as one fourth of the sound speed (c ~ 1), whereas the material velocity (of the order of 0.01) is almost negligible compared with assumed grid velocity (v = 0.25). The analytical and numerical solutions of this problem are superimposed in the Figure 1. Only the spatial interval 24 < x < 34 is shown in order to closely examine the wave fronts. As may be observed, the implemented transport technique captures well the discontinuities of the analytical solution. Performing correctly the convection of all main variables, the algorithm introduces no high frequency damping because no average operation is used. That provides the good conservation properties of the presented algorithm. We can also note that the algorithm is almost insensitive to the grid motion because very close results are obtained with the fixed grid. To reduce numerical oscillations due to the centred representation of the propagation phenomenon in the Lagrangian phase the artificial damping is used.

Figure 1. Liu's stress wave propagation test, axx profile 3.2 Taylor bar impact problem To validate our mixed FE-FV algorithm in multidimensional case we consider a non-linear solid mechanics problem of a cylinder impact against a rigid wall known as Taylor bar impact problem. This test is typically used to validate fast-transient dy-

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namics computer codes because the final length and shape of the cylinder are very sensitive to the constitutive behaviour of its material. The geometrical and material data of the problem are shown in the Figure 2. The cylinder impacts a rigid, frictionless wall with an initial velocity of 227 m/s. The material is supposed elasto-plastic with isotropic hardening. The final time of the simulation is 80/^s corresponding to a moment when all kinetic energy of cylinder is consumed by the plastic deformation process. As the original problem is axisymmetric we discretise only a quarter of the

Figure 2. Taylor cylinder bar impact cylinder using 1050 hexahedrons (with 50 elements in the length). Because of a large elasto-plastic material flow this test is particularly severe for the purely Lagrangian codes. Due to a fast impact velocity, the computational grid, when attached to the material particles, undergoes so strong local compression that element volumes become negative and calculations numerically "explode". In order to achieve the Lagrangian calculation we use a coarse grid near the impact zone. However, when we use the ALE formulation with a regularisation condition, the computational grid is automatically maintained uniform during all calculation. The Figure 3 shows the evolution of the computational grid in Lagrangian and ALE cases. In spite of a great difference between the grid and material velocities in ALE case, involving a high non uniform convective velocity, the algorithm manages successfully with the transport phase. The values of the final height HF and radius Rf of the cylinder obtained by our mixed transport technique are in good agreement with the results computed by various 2D and 3D codes, as can be seen in the following table. Author [HAL 86] [LIU 86] [CAS 95] present paper

Code DYNA-3D PLEXIS-3C DYRAC++

Formulation ALE ALE Lagr. ALE

Solution 3D axisym. axisym. 3D 3D

Hf (mm) 21.47 21.53 21.47 21.45 21.43

Rf (mm) 7.03 6.87 7.14 7.09 7.11

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Figure 3. Comparison of the Lagrangien (left) and ALE (right) calculations 4. Conclusion A mixed ALE algorithm using Finite Element and Finite Volume techniques has been presented and tested on solid mechanics problems with non-linear material behaviour. One can note that this algorithm initially proposed for simulation of hydrodynamic problems dealing with compressible fluids is able to treat fast solid dynamic problems. Capable of dealing separately with compressible fluids and nonlinear solids, the presented approach will be applied to the numerical study of strong fluid-structure interaction problems. Bibliography [HUE 95] A.HUERTA, F.CASADEI and J.DONEA, ALE Stress Update in Transient Plasticity Problems, 4th International Conference on Computational Plasticity - Fun damentals and Applications, Barcelona, 3-6 April, 1995. [POT 97] S.POTAPOV, A Fast Dynamics ALE algorithm based on a mixed Finite Element Finite Volume approach, Ph.D. Thesis, Ecole Centrale Paris, 1997. [NKO 94] B.NKONGA and H.GUILLARD, Godunov type method on non-structured meshes for three-dimensional moving boundary problems, Comp. Meths. Appl. Mech. Engrg. 113, 1994, p. 183-204. [HAL 86] J.O.HALLQUIST and D.J.BENSON, DYNA-3D : User's Manual, Univ. Of California, Lawrence Livermore National Laboratory, Report UCID-19592, 1986. [LIU 86] W.K.Liu, T.BELYTSCHKO and H.CHANG, An Arbitrary Lagrangian Eulerian Finite Element Method for Path-Dependent Materials, Comp. Meths. Appl. Mech. Engrg. 58, 1986, p.227-246. [CAS 95] F.CASADEI, J.DONEA and A.HUERTA, Arbitrary Lagrangian Eulerian Finite Elements in Non-Linear Fast Transient Continuum Mechanics, JRC, European commission, Report EUR 16327 EN.

An Euler Code that can compute Potential Flow

Mani Rad and Philip Roe Department of Aerospace Engineering The University of Michigan Ann Arbor, Michigan 48109-2140 U.S.A. Centre pour Mathematiques et leurs Applications Ecole Normale Superieure de Cachan, France

ABSTRACT A new approach is taken to the computation of the elliptic part of the Euler equations. In each cell of an unstructured triangular grid, on which the solution is stored at the vertices, the residual is decomposed into purely elliptic and purely hyperbolic contributions. The elliptic part is minimised in a norm suggested by an earlier analysis of linearized potential flow. The minimization is carried out by projecting the direction of steepest descent into a surface on which both the entropy and enthalpy are constant. When the method is applied to subcritical flows for which potential solutions should be obtained, the enthalpy is constant to machine zero, and the entropy is constant to an extremely high degree of accuracy. Key Words: Euler equations, potential flow, fluctuation splitting.

1.

Introduction

This paper contributes further to the development of computational methods for the Euler equations (and eventually for other problems that share their structure) that reflect genuinely multidimensional physics. These methods aim to retain the benefits of a physical aproach to strongly discontinuous flows while avoiding the defects of upwind methods applied to almost incompressible flow. There has recently been considerable success in extending compressible flow

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codes to the low Mach number limit by means of matrix preconditioning. The present paper takes the approach of equation decomposition, which is a very closely related concept. We continue with the analysis in [ROE 99-1] where we showed that the decomposition of the residual into elliptic and hyperbolic parts, and the update procedure are sufficiently simple that explicit formulae can be given for them. This enables the singularities of the procedure, near sonic or stagnation points, to be displayed and percieved harmless. In that paper the update for the elliptic part was performed by a straightforward method of steepest descent. Here we constrain the descent direction so that the elliptic part of the update does not affect the entropy or the enthalpy variables. In this way our Euler code is able to compute with very high precision potential flows for which those variables should be exactly constant.

2.

Forms of the Euler Equations

Different aspects of the Euler equations are most readily expressed by choosing different sets of unknowns. For computing compressible flow, the most fundamental choice is probably the set of conserved quantities

since a weak solution of this form of the equations captures shocks that satisfy the Rankine-Hugoniot conditions. Associated with these variables is the flux tensor There is computational convenience in the parameter vector

where h is the total specific enthalpy (E -\- p ) / p , on account of the property that all components of u, F_ are simply bilinear in terms of z. This allows the construction of local linearizations having conservation properties, both in the one-dimensional [ROE 81] and multi-dimensional [DEC 93] cases. Also important, at least to the present approach, are what have been called [HAY 63] the natural variables

where S is entropy, since it is uniquely in these variables that the Euler equations can be decoupled into maximally independent subsystems [HAY 63][ROE 96-1][ROE 96-3]. This enables the essentially different hyperbolic and elliptic behaviour to be computed independently with the minimum of "crosstalk". In this paper we are particularly concerned with the solution of the elliptic subsystem, which contains only the derivatives of pressure and flow angle.

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Fluctuation Splitting

In this approach [ROE 94] [DEC 95], the computational domain is divided into elements in an unstructured way, with the unknowns stored at its vertices. Iteration toward a steady solution from an in intial guess takes place by computing an average residual for each cell (the fluctuation 4>T in element T), and then changing the current solution at each node of the cell T by an amount proportional to c^T;

where a; is a relaxation factor and the weight aj is a matrix to be determined. Our aim is to create a method of this kind that exploits the different properties of the different sets of unknowns. We represent the solution in terms of the parameter vector, and find the conservative residual by integrating over the cell (of area AT)

where the matrices C •= <9F_ jdz and C — <9F_ fdz are locally constant because of the quadratic property [DEC 93]. Our aim is then to reduce this residual in some suitable norm. The norm is chosen by splitting the residual into its elliptic and hyperbolic parts, which is uniquely possible in the natural variables [HAY 63] [ROE 96-1]

where s,n are streamline and normal coordinates. The first two equations represent the elliptic part and involve only the two unknowns of pressure and flow direction; the second pair are the hyperbolic part. In [Roe 99] the elliptic part was solved by minimising that component of the residual, whereas the hyperbolic part was solved as a pair of scalar advection equations. In this paper the elliptic part of the update is constrained not to interfere with the hyperbolic part and vice versa.

4.

The Update Matrix

This gives a convenient way to think about Fluctuation-Splitting schemes. Let the initial state of a cell (in two dimensions) whose vertices are (a, b, c) be

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represented by the twelve scalar quantities

where w is whatever quantity we have decided to store at the vertices. The update can be represented as a matrix multiplication

where U is some matrix that is constant within the cell and u; is a relaxation constant. For example, suppose we intend to update the solution by a steepest-descent minimisation of

where Q is some symmetric matrix representing a local norm. If we write Q = ptp this is equivalent to an LI minimization of P4>T', which is some weighted combination of the residuals. We will treat <j)T as a linear function RTWT of the vertex values because it arises from a linear process to find the local derivatives, followed by multiplication by matrices that are frozen during the update. In general R is a rectangular matrix formed from three 4 x 4 blocks (see the examples below). Hence the quantity to be minimized is

The gradient of this is

and so the update procedure within each triangle

where W is some global constant, implements the required procedure. If we had chosen to store the natural variables x it is easy to show that the four residuals 0X in (7) are given by

where, using standard methods to evaluate the derivatives,

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with u nj = [u(Aj,)j - v(Ax)j] and u sj = [t>(A y )j + u(&x)j] with ( ( & x ) j , (A y )j) the vector representing the side opposite vertex j. Thus s,n now represent directions along and normal to an edge. The first two components of this vector comprise the elliptic part of the problem, if M < 1, so the quantity to be minimized is where where the relative weighting of the continuity and vorticity residuals follows [ROE 96-31. Then the update matrix is

Since in fact we are storing the parameter vector z we must estimate the quantity to be minimised as

and the update becomes

This was the process studied in [ROE 99-1], merely as a first step. It has the disadvantage that minimising the elliptic part of the residuals changes the convected quantities, entropy and enthalpy. We replace it here by a constrained minimization, in which the elliptic part of the residual is reduced as quickly as possible subject to the constraint that this part of the procedure should not change the convected quantities.

5.

Constrained Update Procedure Following [ROE 99] the unconstrained update matrix can be written

where from now on the matrices R incorporate the change of variable dx/dz For example Ra is the matrix

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where

and

After exploiting some coincidences each block of the update matrix becomes

Since we intend complete decoupling at the update level between the elliptic and hyperbolic parts, the elliptic part of the residual should only contribute to changes in pressure and flow angle and not to entropy and enthalpy. Conversely the entropy and enthalpy residuals should change only the entropy and enthalpy. For each natural variable, the direction of a vector along which only that variable changes is

An arbitrary change in the parameter vector dzu coming from the unconstrained minimisation can be expressed in this basis by solving dzu = Pa where a = (as, c*/,, ap, QeY is a vector of coefficients and P = [rs r^ rp TQ\ is the direction matrix. The constrained changes due to the elliptic part are then found by supressing as,h- This is to say,

where the overbar indicates taking the last two components. The constrained update matrix is therefore

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and carrying out the matrix multiplications reveals

The unconstrained steepest descent has the property that changes due to any triangle sum to zero, thereby ensuring conservation. To preserve this property it is necessary to project each triangle residual onto some local set of directions P and then use these for the changes induced by that triangle at each of its nodes. Close to M = 0 these directions degenerate; rs, r^ and rp become nearly parallel. Under these circumstances we found that the code would not converge. To achieve convergence we needed to use a nonconservative version in which the changes at each node were projected onto the allowable directions. However, at higher Mach numbers the conservative technique worked well. 6.

Computational Examples

The first figure presented is a potential flow calculation on a symmetrical airfoil at zero incidence, where the least-squares method is used only in the subsonic region. Where a supercritical region exists, the hyperbolic part of the problem is handled using the PSI advection scheme. No special procedure was used to match the calculations across the sonic line or across the shock. The last two figures show Euler computations on a cylinder in a subcritical case (Moo = 0.35) and in the incompressible limit (Moo = 0.01) using the residual decomposition scheme described in this paper. The elliptic part of the problem was handled using the constrained least squares scheme as presented in sections 3,4 and 5 while the LDA upwind advection scheme was used on the hyperbolic part. Results shown are for a grid consisting of about 3544 cells and it can be seen that a very small amount of spurious entropy is generated very close to the cylinder wall and especially in the stagnation regions. The rate of entropy generation is measured by

and is measured to be 1.0 x 10 9 for the incompressible case and 1.0x10 ' for the subcritical case. Not shown in the figures but of certain importance is that

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enthalpy stays everywhere constant and equal to its initial farfield value. These results reveals that the projection ideas discussed in section 5 are completely successful for the enthalpy and satisfying for the entropy. In any case, it seems that the present formulation provides a solid framework for treating sensitive regions around the wall boundary. The accuracy of the scheme was measured by comparing the numerical solution to the analytical solution in the incompressible case. Error is defined as (v is velocity vector)

and was measured to be 9.0 x 10-4 for a coarse grid of 860 cells and 2.3 x 10 -4 for a finer grid of 3544 cells. In other words, the scheme's convergence behavior is very close to second order.

7.

Concluding Remarks

The present contribution is still not yet intended to provide a practical recipe for solving the Euler equations. The main defect is that steepest descent is, by itself and even with the improvements derived here, a very slow way to solve nonlinear equations. And applied to a linear elliptic system it amounts only to a point Jacobi relaxation. However, it forms the starting point for many other (mainly Newton-like) methods that are extremely efficient, and which have the same fixed points. However, the objective here is entirely to find an iterate that has a good fixed point, not one that reaches some fixed point quickly. It also remains to be proved that an analogous method will work in three dimensions. There is no problem formally. All the formulae given here extend straightforwardly. The issue is whether the streamwise vorticity that becomes coupled to the potential flow in three dimensions [ROE 96-2] will be well captured. It is significant that discretisations exist [ROE 99-2] for which correct growth of the discrete vorticity is automatic, in the sense that a discrete Kelvin Theorem is guaranteed. Future work will determine whether this is possible.

8.

References

[DEC 93] DECONINCK, H., ROE, P. L., STRUUS, R. J., <>, Computers and Fluids, 22, p215, 1993. [DEC 95] DECONINCK, H, PAILLERRE, H., ROE, P. L.,

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[HAY 63] HAYES, W, PROBSTEIN, R. F., Hypersonic Flow Theory, Academic Press, 1963. [ROE 81] ROE, P. L., « Approximate Riemann solvers, parameter vectors and difference schemes », J. Comput. Phys., 43, pp357-372, 1981. [ROE 94] ROE, P. L., « Multidimensional upwinding, motivation and concepts », von Karman Institute Lecture Series 1994-04 [ROE 96-1] ROE, P. L., MESAROS, L. M., ^Solving steady mixed conservation laws by elliptic/hyperbolic splitting », 13th International Conference on Numerical Methods in Fluid Dynamics, Monterey, July, 1996. [ROE 96-2] ROE, P. L., TuRKEL E.,

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Figure 1. Subsonic (M Mach contours.

= 0.36) and transonic (M

= 0.85) NACA 0012,

Figure 2. Cylinder flow in the incompressible limit (M entropy contours. Euler solution.

= 0.01), pressure and

Figure 3. Cylinder flow in the subcritical range (M entropy contours. Euler solution.

— 0.35), Mach and

Finite volume evolution Galerkin methods for multidimensional hyperbolic problems M. Lukacova-MedvicFova 13 , K. W. Morton 2 , G. Warnecke1

l

lnstitut filr Analysis und Numerik, Otto-von-Guericke-Universitdt Magdeburg, PSF 4120, 39 106 Magdeburg, Germany, 2 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom (also Oxford University Computing Laboratory), 3 Institute of Mathematics, Faculty of Mechanical Engineering, Technical University Brno, Technickd 2, 61639 Brno, Czech Republic

ABSTRACT The finite volume evolution Galerkin method couples a finite volume formulation with an approximate evolution Galerkin operator, which takes into account all of the infinitely many directions of propagation of bicharacteristics for multidimensional systems. Piecewise linear recovery yields second order accuracy even with the first order approximate evolution operator. Numerical comparisons of the evolution Galerkin schemes with the commonly used finite volume methods for the wave equation system and for the Euler equations are presented. Key Words: genuinely multidimensional schemes, hyperbolic systems, wave equation, Euler equations, evolution Galerkin schemes, finite volume methods

1. Introduction

It is our belief that the most satisfying methods for approximating evolutionary PDE's are based on approximating the corresponding evolutionary operator. In order to construct a genuinely multidimensional numerical scheme for hyperbolic conservation laws all of the infinitely many directions of propagation of bicharacteristics have to be taken into account.

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This is the main idea of the evolution Galerkin (EG) methods, which evolve the initial data using the bicharacteristic cone and then project them onto a finite element space. In the recent paper Lukacova, Morton and Warnecke [LMW 99] three new first order evolution Galerkin schemes for a system of hyperbolic equations, and particularly for the wave equation system are derived and analysed. It is shown that the evolution Galerkin scheme (denoted in the paper [LMW 99] as the EG3 method), which is based on the general theory of bicharacteristics for hyperbolic conservation laws gives the most accurate numerical scheme. This is a base for the second order scheme, which will be constructed using a piecewise linear recovery. In the recent years the most commonly used numerical shemes for hyperbolic conservation laws were the finite volume methods, which are based on a type of directional splitting and on an approximate solution of one-dimensional Riemann problems. Their popularity is particularly due to the simplicity of their formulation as well as implementation. However, it is a known fact that these methods can produce structural deficiences in the solution for some special multidimensional problems (see, e.g., [FEY 92], [LEV 97], [LMW 98], [LMW 99]). The finite volume evolution Galerkin method combines advantages of both approaches: the simplicity of the finite volume formulation and the multidimensionality of the evolution Galerkin schemes. 2. Finite volume evolution Galerkin methods

Consider a general hyperbolic system in d space dimensions

where l , . . . , d represent given flux functions and the unknown functions are U = . Let us denote by E ( s ) : (Hk the exact evolution operator associated with a time step s for the system (1), i.e.

We suppose that Sph is a finite element space consisting of piecewise polynomials of order p 0. Let be an approximation in the space S^ to the exact solution at a time tn > 0 and take Er : Srh -> (Hk(IRd))m to be a suitable approximation to the exact evolution operator E(r), r 0. We denote by Rh : S a reconstruction operator, r > p > 0 . In the present paper we

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shall limit our consideration to cases of constant time step A/, i.e. tn = r?A/, and of a uniform mesh consisting of d-dimensional cubes with a uniform mesh size h. Definition 1. Starting from some initial value U0_ at time t= 0, the finite volume evolution Galerkin method (FVEG) is recursively defined by means of

where the central difference v(x -f h/2) — v(x — h/2) is denoted by STv(x) and represents an approximation to the edge flux difference. The cell boundary value is evolved using the approximate evolution operator ET to tn + T and averaged over 0 < r < At and along the cell boundary, i.e.

where - is the characteristic function of . There are several advantages to this formulation. The most important is that the first order accurate approximation E T to the evolution operator E ( r ) yields an overall second order update from U_n to U_n+ l. To obtain this second order approximation in the discrete scheme it is only necessary to carry out a recovery stage at each time level to generate a piecewise linear approximation U = RhU_n from the piecewise constant to feed into the calculation of the fluxes. In the next section we illustrate this procedure for the wave equation system in two space dimensions. 3. Wave equation system

The wave equation can be written down as a first order hyperbolic system

with the unknown functions . Consider a cone with the apex P — (x, y, t-\At) and the base points Q = Q(0) = (x+cAt cos 0, y+cAt sin 0, t) parametrized by the angle 6 £ [0, 2]. Denote by P' = (x, y, t) the center of the base of the cone. The lines from Q(0] to P generating the mantle of the so-called bicharacteristic cone are called bicharacteristics, see, e.g., [LMW 99] for more details. Using the theory of bicharacteristics it can be shown that the solution (, w, v) at the point P is determined by its values on the base as well as on the mantle of the characteristic cone and the exact evolution formulae can be derived. In the

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recent papers Lukacova, Morton and Warnecke [LMW 98], [LMW 99] several approximate evolution operator for the wave equation system were analysed. It was shown that the following approximate evolution operator leads to the best first order scheme in terms of accuracy, see [LMW 98], [LMW 99]. 3.1. First order approximate evolution operator

Denote by Ph L2 - projection onto a space of piecewise constant functions in IR2, then we obtain the first order scheme which is in [LMW 99] referred to as the EG3 scheme. Space integrals coming from the projection step are computed exactly, i.e. no numerical quadrature is used. The finite difference formulation can be found in [LMW 99], where the coefficients of the scheme are given explicitely. 3.2. Second order reconstruction In order to construct the second order FVEG scheme we take the first order accurate approximate evolution operator (6) - (8) and define a bilinear reconstruction Rh- There are many possible recovery schemes, which could be used. For our computation we choose a discontinous bilinear recovery using a four point averages at each vertex, but others can be used as well. It is taken to be conservative and given as

where tation is used for averaged value

an analogous nooy. For the computation of fluxes through cell edges the time has to be known, see (4). Instead of exact time integration

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the second order midpoint rule is used, i.e.

Two dimensional space integrals of bilinear functions Rh with respect to 9 and S which occur in (9) are computed exactly without any numerical quadrature and thus all of the infinitely many directions of propagation of flow information are taken explicitely into account. The above construction leads to the overall second order scheme, which gives in regions of smooth solution very accurate results even on coarse grids, see Table 1 below. 3.3. Numerical results We consider the space periodic problem for the wave equation system (5) with the initial data

In this case the exact solution can be easily computed [LMW 99]. In Table 1 we compare errors of several second order schemes, namely the second order FVEG method, the Lax-Wendroff method (LW) and the standart finite volume fluxvector directional splitting method (FV-FVS), which uses a MUSCL technique for the flux computation and a second order Runge-Kutta time approximation. For the details on the latter one see, e.g., [KRO 97]. We use meshes of 20 x 20, 40 x 4 0 , . . . , 640 x 640 cells and compute also the experimental order of convergence (EOC) from two meshes of sizes NI and N2 as

In all cases the results are for a CFL-number v of 0.45 and an end time of T = 0.2. Experiments for several other values of v and T confirm the second order accuracy of the FVEG scheme. N / \\LL(T)-Un\\

20 40 80 160 320 640 EOC

FVEG 0.008647 0.001789 0.000925 0.000290 0.000080 0.000021 1.94

LW 0.065500 0.016472 0.004133 0.001033 0.000258 0.000064 1.99

FV-FVS 0.057748 0.014502 0.003617 0.000905 0.000227 0.000057 2.00

Table 1. Comparison of accuracy of the FVEG method, the Lax-Wendroff method and the FV flux-vector splitting method

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4. Euler equations

The system of Euler equations describing the motion of compressible flow in two space dimensions can be written in the form of hyperbolic system (1) with d — 2; the definition of U_ and Fi(LO is well-known and can be found e.g. in [KRO 97]. If R = (rj n, r 2 i n , £3n, r_4n) denotes the matrix of right eigenvectors we can decompose the vector of conservative variables U_ for any direction n in the following way

A general procedure for derivation of the exact evolution operator for hyperbolic problems can also be applied to the Euler equations. For linear problems this procedure works with the characteristic variables W = R~ U_. Now, instead of W the vector a is used. The following approximate evolution operator for the Euler equation was derived in [OST 97]

where O denotes the unit sphere, is a local speed of sound and u represents the fluid velocity. After L2 - projection onto a space of piecewise constants we obtain the EG scheme for the Euler equations. It was shown by Ostkamp [OST 97] that Fey's method of transport [FEY 92] for the Euler equations can be reinterpreted as the above evolution Galerkin scheme. 4.1 Numerical results We take the well-known test problem, namely the two-dimensional Sod's problem and compare the behaviour of the evolution Galerkin method (11) and the LeVeque wave propagation algorithm [LEV 97], which is available as a public domain software package called CLAWPACK. In order to avoid a discussion of the limiters we compare here only first order schemes. Note however that in LeVeque's wave propagation algorithm also first order correction terms for xy-cross derivative are included. The computational domain is the square [-1,1] x [-1,1]. To ensure the CFL stability condition, the CFL number is taken 0.8. We choose periodic boundary conditions and the following initial data

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Solutions at time T = 0.2 are computed on a quadrilateral grid with 200 x 200 grid cells. In Figures 1 and 2 the isolines of solution obtained by the evolution Galerkin method (11) and by the wave propagation algorithm, respectively are drawn. The {/-velocity is not depicted since it is symmetric to the x-velocity. For the evolution Galerkin scheme the resolution of the flow phenomena is the same in all directions and information is moving in infinitely many directions in a circular manner. However, we can notice that the wave propagation algorithm, which makes use of an improved directional splitting, does not preserve circular symmetry in such a good manner as our scheme and some dependence of the solution on the grid can well be seen.

Figure 1. Evolution Galerkin scheme: density (left), x-velocity (middle), pressure (right)

Figure 2. Wave propagation algorithm: density (left), x-velocity (middle), pressure (right)

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5. Acknowledgement

This research has been supported under the DFG Grant No. Wa 633/6-1 of Deutsche Forschungsgemeinschaft, and partially by the Grant No. 201/97/0153 of the Czech Grant Agency and by the DAAD. 6. Bibliography

[FEY 92]

FEY M., JELTSCH R., «A simple multidimensional Euler schemes, Proceedings of EC COM A S'92, Elsevier Science Publishers, Amsterdam, 1992, p. 667-671.

[KRO 97]

KRONER D., Numerical Schemes for Conservation Laws, Wiley - Teubner, Stuttgart, 1997.

[LEV 97]

LEVEQUE R.J., «Wave propagation algorithms for multidimensional hyperbolic systems», J. Comp.Phys., 131, 1997, p. 327-353.

[LMW 98]

LUKACOVA - MEDVIDOVA M., MORTON K.W., WARNECKE G., «On the evolution Galerkin method for solving multidimensional hyperbolic systems», Proceedings of ENUMATH'98, World Scientific Publishing Company, Singapore, 1998, p. 445-452.

[LMW 99]

LUKACOVA - MEDVIDOVA M., MORTON K.W., WARNECKE G., Evolution Galerkin methods for hyperbolic systems in two space dimensions^, Preprint University of Bath, England, 1999, submitted to MathComp.

[OST 97]

OSTKAMP S., «Multidimensional characterisitic Galerkin schemes and evolution operators for hyperbolic systems, Math. Meth. Appl Sci., 20, 1997, p. 1111-1125.

Nonlinear anisotropic artificial dissipation Characteristic filters for computation of the Euler equations Thorsten Grahs, Andreas Meister and Thomas Sonar Institut fur Angewandte Mathematik Universitdt Hamburg, Bundesstr. 55 20146 Hamburg, Germany ABSTRACT We employ a nonlinear anisotropic diffusion operator like the ones used as a means of filtering and edge enhancement in image processing, in numerical methods for conservation laws. It turns out that algorithms currently used in image processing are very well suited for the design of nonlinear higher-order dissipative terms. In particular we stabalize a central scheme, known for its oscillating behaviour by the construction of a nonlinear diffusion term. Using information from the data a so called structure tensor following from ideas of anisotropic diffusion in image processing due to Weickert is constructed, containing information about the orientation and strength of the necessary diffusion. In particular this means constructing a diffusion matrix consisting of eigenvectors parallel and perpendicular to discontinuities and eigenvalues denoting the amount of dissipation depending on the local strength of the gradients. These directions are used to steer the amount of dissipation which means surpressing diffusion across the shock front and using the perpendicular direction to enable the necessary diffusion to stabilize the underlying second order scheme. The diffusion terms are based on the artificial compression method (ACM) due to Harten and the extensions of Yee. The limiting takes place on characteristic based diffusion terms acting on the characteristic velocity. Numerical results are shown for the two-dimensional Euler-equations. Keywords: low dissipation, central finite difference schemes, shock-capturing methods, anisotropic diffusion, characteristic filters, Euler equations

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1. Introduction

The construction of suitable artificial viscosity terms for stabilizing finite volume and finite difference schemes of higher order is a difficult task. In the last decade we observed therfore a strong tendency to construct numerical approximations of conservation laws without explicit knowledge of their numerical diffusion. The modern total variation diminishing (TVD) or essentially non-oscillatory (ENO) shemes belong to this class, in which a basic firstorder scheme is improved by the use of sophisticated recovery functions, see [Son97, MS99]. There, are however, certain circumstances in which an approach using explicit construction of artificial dissipation would be advantageous. If we consider pseudospectral methods the concept of ENO recovery is very hard to apply if the degree of the polynomials used is high. Here one would like to compute shocked solutions with central schemes and post-process the numerical solution such that high frequency oscillations are filtered, and shocks are steepend and represented with high resolution. Over the years there were no general attempts to derive a constructive theory which would enable the design of suitable artificial viscosities within the CFD community. However, filtering and edge enhancement is a fundamental task in image processing and in recent years a theory of nonlinear anisotropic diffusion was created and can now be found in textbooks like [Wei98]. In a noisy picture one also would like to filter the high frequency components before detecting the edges (i.e. jumps in grey level). Then one would like to enhance the edges in order to represent the edges in high resolution. Now there is nothing which keeps us from interpreting our numerical solution corresponding to the conservation law as a photograph or picture. In the same way the photographer would very much prefer to see the contours on his picture as sharp thin lines the numerical analyst would prefer to see shocks as crispy lines instead smeared thick regions. To accomplish this, the picture as well as the numerical solution have to be denoised. After removing the high frequencies we would like to spend a dose of diffusion tangential to shocks - that is what anisotropy is all about in this context. In contrast, in the vicinity of shocks we would like to solve a kind of nonlinear anisotropic backward heat equation to enhance the structure of a shock front. Devices and algorithms satisfying exactly these requirements are ready to use if one is willing to enter the area of image processing. The aim of this paper is mainly to show how one can use ideas from image processing in the area of the numerical treatment of conservation laws. We have used suchs methods succesfully for scalar mixed Burgers-advectionequation namely stabilizing a Lax-Wendroff scheme and surpressing the oscillatory behaviour of the scheme in the vicinity of shocks (see [GMS98]). Here we would like to discuss the extension to the Euler equations. First we describe the governing equations and the basic scheme. In the following we construct the so called structure tensor to gain informations about the local strength and orientation of the shock. This information will be used to develop a diffusion

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matrix to steer the amount of diffusion necessary to stabilize a second order central scheme.

2. Anisotropic diffusion applied to characteristic filters 2.1.

Governing Equations

We consider the two dimensional time-dependent Euler equations for a compressible gas in equilibrium in conservation form, i.e.

in which

denote the vector of conserved quantities and the flux functions in x and y, respectively. If t denotes time and x = ( x 1 , x t } T 6 space coordinates, the mapping

denote density, velocity, pressure, total energy and enthalphy, respectively. Enthalphy is defined by

To close the system, an equation of state is needed. For ideal gases one uses

where 7 denotes the ratio of specific heats. In the case of dry air one assumes a value of 7 = 1.4.

2.2.

Construction of the characteristic filter

We discretize on a cartesian grid with mesh size Ax, Ay, respectively. Since in this case finite volume and finite difference techniques are equivalent, we use in the following a typical finite difference notation. We start from an explicit central scheme written as

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where the numerical flux functions include the filter terms and hence the cross diffusion parts and can be written as follows:

The filter operators LA, LB depend on the Jacobian-matrix A and B of the flux function F and G, respectively. The filter terms Lc models some kind of cross diffusion and depends on a combination of the Jacobians, namely

(see [Hir90] for details). The vector n : = (nx'1,nxl)T is chosen in direction of the cell diagonals, i.e n+ := (l,l) T /\/2 and n~ :~ ( 1 , — l) T /\/2- The matrices are all evaluated as a Roe average ( see [RoeSl]). So one writes the corresponding filter terms as

and similar for the other terms. Here R^'> denotes the matrix of the right eigenvectors corresponding to the appropriate matrices written as superscript. <$(') denotes the real filter term. In the basic construction, we follow the recent paper of Yee and her co-workers [YSD99], based on the Artificial Compression Method (ACM) of Harten and the extensions of Yee (see f.e.[Har78, Har83, Yee85]). We extend these method by the integration of a directional based - that its what we denoted with anisotropic - diffusion known from image processing. So the elements of 3>('^ in (1) are denoted by ^'\l — 1 , 2 , 3 , 4 . We give here as an example the construction of the term <j>i+\/2 •• All other terms are constructed in the natural extension of this definition depending on the matrices and grid points were they are evaluated. So the elements of j 1/2 • write as

The denotes the weighting coefficients steering the anisotropic diffusion. The construction of this coefficient will be described in the following section. The

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choice of the is exactly the same like the one described in [YSD99] with the use of the limiter functions gl- as

Additional possible choices for this function and a detailt description are given in [YSD99]. The definition of will be give below. The choice of the filter function depends on the characteristic splitting of as the flux representation of the gradients. Thus, we consider elements The different a depend again on different matrices of the right eigenvectors depending on the gridpoint were they are evaluated. They are given as:

(2)

Hence oli+1/21j+1/2corresponds to a1i+1/22j+ 1/2 and so on.... The filter function writes as (3)

(4)

The a' + 1 / 2 in (3),(4) are the characteristic speeds of the corresponding Jacobians, which are equal to their eigenvalues. This term can also be weighted with the diffusion coefficient /3li+1,2 . steering the amount of dissipation corresponding to this direction.

2.3.

The structure tensor

We start from the so called structure tensor

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smoothed version of the characteristic gradients R 1 AC/,5, where U$ is a presmoothed version of the data Un. This means convolution with a Gaussian kernel whith convolution scale 6, i.e

Since in the continnous case this is equivalent with solving the heat equation we apply this to the data Uij with stopping time T = ^S2, where S is a parameter which has to be chosen. In order to remove the small scale oscillations by means of this smoothing technique we define S depending on the grid size h := ^/NxKy. Consequently, the structure tensor reads as (5)

with

where a1''5 corresponds to the smoothed data U$. Therbye one can also use a smoothed version of the structure tensor (5), i.e (6)

which means component-wise convolution with scale v, which denotes the width of the averaging region. In practice we are solving the heat equation for each component seperately.

2.4.

The diffusion

matrix

After having constructed the structure tensor we are going to compute the eigenvectors and eigenvalues of (6), (7)

(8)

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Therbye the corresponding eigenvalues are given by (9)

Now we construct the dissipation matrix D by introducing the ansatz due to Weickert [Wei98]:

(10) Here Vv denotes the matrix of the eigenvectors (7), (8) and A,, = diag(/i,/2) represents a diagonal matrix where the diagonal elements have to be calculated in a convenient manner as described below. In order to recover shocks (or, equivalently, in order to enhance edges) the diffusivity l\ perpendicular to edges should be reduced if the contrast \\^ is high. This can be achived by an anisotropic regularization of the Perona-Malik model [PM90] also adapted from Weickert:

The values of m and Cm are chosen in such a way that the so-called flux 4>(s) := s$(s) is increasing in an interval s 6 [0, A] and decreasing in s g]A, oof. The choices depend on a one-dimensional analysis of the Perona-Malik model. In agreement with Weickert we chose m — 4 and thus €4 — 3.31488. The so-called contrast parameter A, separating areas with forward (low contrast) from backward (high constrast) diffusion, can be chosen freely. Based on numerical experiments it turns out that for calculations concerning systems it is usefull to have an adaptive parameter instead of a fixed one. We calculate for each variable the maximum of the according gradients in every timestep and choose a fixed percent of this maximum to determine the parameter A. Thus, the diffusion matrix (10) can be written as

(11) with coefficients

where V\.}P — (VII->P, ^i 2 i p) T - The superscript / reminds that the diffussion matrix (10) and so the structure tensor (5) resp. (6) have to be computed for every characteristic variable seperatly.

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Weickert proved that there is always a finite difference stencil which leads to a stable scheme. Moreover, he was able to show that three directions suffice to guarantee a non-negative discretization, since a negative discretization is equivalent to an ill-posed problem. The discretization of a 3 x 3 stencil reads as follows:

This gives the weighting coefficients for the dissipative fluxes which leads to a steering of the dissipation terms depending on the magnitude of the gradients as described it in the last section.

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3. Numerical results As an example we use a test case given by LeVeque [LeV93] with initial data

and Ui = (2, 0, 0,15) T , Ur = (1,0,0,1) T in primitive variables. The solution consist of a shock running outwards followed by a rarefaction wave and a contact discontinuity. A second shock moves inwards towards the center. We use an equidistant discretization with Ax = Ay = h = 0.025 and a CFL-Number = 0.4. The smoothing parameters are given with 8 = 0.25/J. and v = 0.0 which means no smoothing of the structure tensor takes place. The contrast parameter A is chosen as 0.4max(Ai ; i x ).

Figure 1: Solution of the test case (12) at time t=0.13

4. Concluison We have extended the characteristic filter approach by Yee, Sandham and Djomehri [YSD99] by an anisotropic directional based diffusion from image processing. Since this integration into the field of numerical conservation laws is a novel approach, the calculations presented here are preliminary results and will need further research. Overall we are quite optimistic concerning the behaviour of the suggested extension of this scheme.

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References [GMS98] Th. Grabs, A. Meister, and Th. Sonar. Image Processing for Numerical Approximations of Conservation Laws: Nonlinear anisotropic artificial dissipation. Hamburger Beitrage zur Angewandten Mathematik, Reihe F: Computational Fluid Dynamics and Data Analysis 8, 1998. (submitted to SI AM J. Sci. Comp.). [Har78]

A. Harten. The artificial compression method for computation of shocks and contact discontinuities. III. self-adjusting hybrid schemes. Math. Comp., 32:363-389, 1978.

[Har83]

A. Harten. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 49:357-393, 1983.

[Hir90]

Ch. Hirsch. Numerical computation of internal and external flow, volume 2. J. Wiley fe sons, 1990.

[LeV93] R.J. LeVeque. Simplified multi-dimensional flux limiter methods. In M.J. Baines and K.W. Morton, editors, Numerical Methods for Fluid Dynamics 4, pages 175-190. Oxford University Press, 1993. [MS99]

A. Meister and Th. Sonar. Finite Volume Schemes for compressible fluid flow, in press: Surveys of Mathematics in Industry, 1999.

[PM90]

P. Perona and J. Malik. Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach.Intell, 12:629-639, 1990.

[RoeSl]

P.L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43:357-372, 1981.

[Son97]

Th. Sonar. Mehrdimensionale ENO-Verfahren. Advances in Numerical Mathematics. B.G.Teubner Stuttgart, 1997.

[Wei98]

J. Weickert. Anisotropic Diffusion ner, Stuttgart, 1998.

[Yee85]

H. C. Yee. Construction of explicit and implicit symmetric TVD schemes and their application. J. Comput. Phys., 68:151, 1985.

in Image Processing. B.G. Teub-

[YSD99] H. C. Yee, N. D. Sandham, and M. J. Djomehril. Low-dissipative high-order shock-capturing methods using characteristic-based niters. J. Comput. Phys., 150:199-238, 1999.

Nonlinear projection methods for multi-entropies Navier-Stokes systems

Christophe BERTHON

Frederic COQUEL

ONERA,

LAN-CNRS,

BP 72, 92322 Chdtillon Cedex, FRANCE.

4> place jussieu, 75252 Paris Cedex 05, FRANCE.

ABSTRACT This paper is devoted to the numerical approximation of the compressible Navier-Stokes equations with several independent pressures. Several models derived in plasma physics or in turbulence typically enter the proposed framework. The striking novelty over the usual Navier-Stokes equations stems from the impossibility to recast equivalently the present system in full conservation form. Classical finite volume methods are shown to fail in the capture of shock layers. We propose a new method, the so-called nonlinear projection operator, for correcting the errors while preserving all the stability properties. Key Words: Navier-Stokes equations, Non conservative products, Nonlinear projection.

1. Introduction The present work treats the numerical approximation of the solutions of the Navier-Stokes equations for a compressible fluid modelled by two independent pressures, e.g. each of the pressures comes with its own specific entropy. Despites that such models are seen to exhibit several close relationships with the usual Navier-Stokes system, the fundamental discrepancy stays in the lack of an admissible change of variables that recasts the system in full conservation form. None of the entropy balance equations boils down to a conservation law involving non conservative products that account for dissipative phenomena : namely the entropy dissipation rates. Such systems occur in several distinct physical settings. They arise for instance in plasma physics and they can be also recognized within the frame

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of the "two transport" equations models for turbulent compressible flows. All these models are addressed below with the assumption of a large Reynolds number. The numerical capture of the viscous shock layers is of primary importance in the present work. Since the Reynolds numbers of interest are large, these layers display the character of a shock wave in that they differ from their end states only in a small interval of rapid transition. Our purpose is to correctly capture the two end states, VL and VR, of a given shock layer together with its speed of propagation a whitout resolving sharply the viscous layer itself. It is quite well-known that such an issue does not raise special difficulties within the standard frame of the Navier-Stokes equations, e.g. in conservation form. At the very core of this success, is the definition of v#(cr; v^) stays free from the entropy dissipation rate and in this sense, its numerical capture stays also free from the rate of numerical dissipation. The situation turns out to be completely different in the setting of the extended Navier-Stokes equations. Its non conservation form makes this time the triple (cr; VL, v#) to heavily depend on the precise shape of the diffusive tensor. Such a dependence stays at the basis of recent works devoted to hyperbolic systems involving non conservative products (see Lefloch [6], Dal Maso-LeFlochMurat [2], Raviart-Sainsaulieu [8]). In the setting of numerical methods, this dependence implies that the numerical viscous tensor must fit the exact last one. The negative consequences can be found in the numerical result presented below. In the present work, we propose an approach based on the analysis of the discrete dissipation rates of a given Lax entropy. It turns out that the end states require for their correct capture to prescribe explicitely the Lax entropy dissipation rate. This requirement asks the numerical methods to satisfy in turn an imposed rate of entropy dissipation. This non standard issue is precisely the main motivation of the present work. Let us underline that the results we state below extend in a straightforward way to higher space dimensions, taking advantage of the rotational in variance of the equations. 2. Mathematical model

We consider a gas with density p and velocity u, which is modelled by two independent pressure laws p and pT, associated with two constant adiabatic exponents 7 > 1 and 7r > 1. The system of PDE's that governs such a fluid model writes :

(1)

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where the involved temperatures respectively read T — p/'p and TT — pT/pThis convective-diffusive system can be understood as an extension of the standard Navier-Stokes equations when considering an additional PDE for governing an additional pressure. Depending on the closure relations for the viscosities /i, [ir and the thermal conductivities K and KT, several distinct physical models enter the present framework. In this section, all these transport coefficients are assumed to be fixed positive constants for the sake of simplificity in the discussion. The smooth solutions of system (1) obey additional governing equations as we now state : Lemma 1 Smooth solutions of (1) satisfy the following conservation law: (2)

where the total energy pE is defined by pE — ^^—I—^y H—^~[- Smooth solutions satisfy in addition the following balance equations : (3) (4)

where the specific entropies are respectively given by s Consequently, smooth solutions of (1) obey :

where the right hand side follows under the assumption of two constant viscosities p, and IJLT . The three balance equations (2), (3) and (4) can be proved to be the only non trivial additional equations for smooth solutions. As a consequence the discrepancy stays in the lack of four non trivial conservation laws. Indeed, none of the equations (3), (4) and (5) boils down to a conservation law and (1) cannot be recast in full conservation form. After the works by LeFloch [6] and Raviart-Sainsaulieu [8], the non conservation form met by (1) makes the end states of shock layers to depend on the closure relations for the coefficients //, \JLT and /c, KT. In order to assess this issue, let us focuse on the non standard balance equation (5) where by contrast with (3) and (4) which dissipation rates are independently imposed, (5) exhibits a compared rate of both the entropy dissipations. The idenity (5) continues to play a major role in a general setting since they encode a generalized jump condition which turns out to play a central role for our numerical purpose :

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Theorem 2 Assume that p,, ^T, K and KT denote positive constants. Then the triple (cr, VL, vR) associated with the resulting dissipative tensor necessarily obeys the jump relations :

(6.i) (6-ii) (6.iii) and necessarily satisfies the two entropy inequalities : (7)

Under an appropriate setting (see the companion paper [I]), (7) can be specified as follow :

(8) defining a generalized jump condition where the involved averages find a unique definition (see [1] for details). Remark 3 In view of the jump relations (6) and (8), one of the two entropies, either ps or psT, must be understood as a nonlinear function of the four remaining independent variables (p,pu,pE,.}.

3. Godunov methods with nonlinear projections For the sake of simplicity , we do not address the discretization and we set K = KT — 0 We refer the reader to the companion paper [1] for the required discrete formulae. 3.1. L2 projections (tn -+ tn+1'~) An equivalent system using the conservation laws for p, pu, pE and the evolution law governing psT is considered. Notice that the evolution law for ps can be understood as an additional law satisfied by the smooth solutions. Choosing consitant formulae, based on a Godunov method for instance, the discrete solution satisfies, formally, the following dissipation rates of entropie :

(9)

(10)

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As a consequence of the error which occurs in (10), the expected generalized jump relation (8) is systematicaly violated. The reader is referred to [1] for a rigourous proof and to the numerical results below for an illustration of the negative consequences of such a failure.

3.2. Nonlinear projection methods (tn+1,-- --- tn+1 ) We propose to add an additional step to classical L2 projection methods, the nonlinear projection step, which purpose is to correct the errors. In order to preserve the required conservation properties, let us define :

Setting vf + i = ( / 9^ I '-,(p W ) l n + i '-,( / 9E)^ 1 '-,(p5 r ) t ni + 1 ), to enforce the validity of the generalized jump condition at the discrete level, we propose to seek for (pST)™+11as a solution of : :

The above nonlinear problem in the unknown (psT}™+1 can be shown to admit a unique solution as soon as the approximate Riemann solver involved in the first step obeys discrete entropy inequalities for the Lax pair (ps,psu). The nonlinear projection step (12) allows to prove in addition the following stability results : Theorem 4 Under the required CFL restriction, the following discrete entropy inequalities are satisfied :

for all strictly decreasing and concave functions <j> and if). The following maximum principles for the specific entropies s and ST are met :

Both the pressures Pni+1 and PTin+1 stay positive as soon as the density pn+1 i s positive.

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The proof of the above statement is detailled in [1].

3.3 Numerical results The ability of the schemes in the capture of shock layers for (1) is evaluated when testing their sensitivity in the prediction of the end states with respect to the mesh refinement. The initial data is made of two constant states prescribe by Test A

7 1.4

7r

t*T/V

1.6

0.01

B

1.4

1.6

1

C

1.4

1.6

100

P 1 1.05518 1 1.92678 1 1.01595

u 1 -0.88895 1 -1.25451 1 -0.93415

P 1 0.15031 1 2.74247 1 0.24142

Pr

0.6 0.33869 0.6 2.09561 0.6 0.15528

Problems are directly motivated by the three distinct regimes that underly the flow model under consideration and that are dictated by the amplitude of the viscosity ratio ^T/^- The viscosities uandd /JLT are assumed to be positive constants we referre to [1] for the setting of varying viscosities and non zero heat conductivities, conductivities. In all the benchmarks discussed below, the Reynolds number is set at the constant value Rey = 105.

All the figures assess that the usual numerical strategy (see [7] or [5] concerning the details of this method) grossly fails to restore the correct end states in the three regimes. Turning considering the L2 projection method, the discrete solutions agree with the exact ones only in case A. Such a property no longer holds for problems B and C and consequently large errors occur. These two schemes furthermore suffer from a dramatic sensitivity with respect to mesh refinements for problem C in that discrete solutions do not seem to converge to a given limit function even for the finest proposed grids. By contrast and concerning benchmarks A and B, the discrete solutions stay non sensitive with respect to the mesh refinement but the "limit" function does not coincide with the expected exact solution. Turning considering the nonlinear L2 projection method, it produces approximate solutions that achieve a fairly good agreement with the exact solutions while staying almost non-sensitive with Ax in the three investigated regimes.

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Classical Scheme

L2 Projection Scheme

Figure 1: Problem A : Classical Scheme

Nonlinear Projection Scheme

ur / u < < < 1

L2 Projection Scheme

Figure 2: Problem B

313

:: ut / u = 1

Nonlinear Projection Scheme

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Figure 3: Problem C : ur / u> > 1 3.4. Bibliography [1]

[2]

C. BERTHON AND F. COQUEL, Nonlinear projection methods for systems in non conservation form, work in preparation.

G. DAL MASO, P. LEFLOCH AND F. MURAT, Definition and weak stability of a non conservative product, J. Math. Pures Appl, 74, 483-548 1995.

[3]

E. GODLEWSKI AND P.A. RAVIART, Hyperbolic systems of conserva-

tions laws, Applied Mathematical Sciences, Vol 118, Springer 1996. [4]

T. Y. Hou AND P. G. LEFLOCH, Why nonconservative schemes converge to wrong solutions : error analysis, Math, of Comp., Vol 62, No 206, 497-530 1994.

[5]

B. LARROUTUROU AND C. OLIVIER, On the numerical appproximation of the K-eps turbulence model for two dimensional compressible flows, INRIA report, No 1526 1991.

[6]

P.G. LEFLOCH, Entropy weak solutions to nonlinear hyperbolic systems under non conservation form. Comm. Part. Diff. Equa. 13, No 6, 669-727 (1988).

[7]

B. MOHAMMADI AND O. PlRONNEAU, Analysis of the K-Epsilon Turbulence Model, Research in Applied Mathematics, Masson Eds 1994.

[8]

P. A. RAVIART AND L. SAINSAULIEU, A nonconservative hyperbolic system modelling spray dynamics. Part 1. Solution of the Riemann problem, Math. Models Methods in App. Sci., 5, No 3, 297-333 1995.

About a Parallel Multidimensional Upwind Solver for LES

D. Caraeni S. Conway L. Fuchs Division of Fluid Mechanics, Lund Institute of Technology SE-221 00 Lund, Sweden

ABSTRACT A parallel flow solver has been developed at Lund Institute of Technology, for Large Eddy Simulations (LES) of turbulent compressible flows. The numerical algorithm is based on the Residual Distribution scheme approach. The scheme employs an extremely compact stencil, while still having second order accuracy, which makes it well suited for parallelization. In this paper we report about the numerical scheme and about the parallel algorithm implemented in the code. First the performance concerning the parallel scalability is addressed. Finally some results for the LES of the classical channel flow are presented. Key Words: Residual Distribution Scheme, Large Eddy Simulations, Parallel Flow-Solver.

1. Introduction The advances made in computer technology over the last years, have led to a great increase in the engineering problems which can be simulated using CFD. The computation of flows over complex geometries at relatively high Reynolds number is becoming more common using LES, as recent reviews on the subject have shown [PIO 98]. LES for engineering applications are typically extremely expensive, requiring huge resources in terms of processor power and memory, and long computational times. When using large parallel computations, LES becomes accessible for many industrial applications. Accurate numerical algorithms, well suited for parallelization are needed in order to make LES available for production CFD.

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2. LES Equations The expression for the LES equations, can be obtained by "top-hat" Favrefiltering of the time dependent compressible Navier Stokes equations, written in conservative form:

The subgrid terms representing the subgrid stress tensor, rs^s = p[uiUj — UiUj],, the subgrid heat flux, Hj9S = ~p[Euj — EUJ], and the subgrid viscous work, GSj9S = ~f>\TijUii — TijUi], all require modelling. In the present simulations, the r^js was modeled by using a Sub-Grid Scale (SGS) model, as described below. All the other SGS terms presented above, are not explicitly modeled. 3. SGS Models In the Large Eddy simulation technique, the largest scales of the turbulence are resolved and the effect the smallest scales have on the resolved scales, is modeled. This is done by introducing a SGS model. In the present code, three SGS models have been implemented: the Smagorinsky model, Lilly's dynamic model [LIL 91] and the Dynamic Divergence model (DDM)..Thehe DDM model is a novel, anisotropic dynamic SGS model with independently determined coefficients in each coordinate direction. Note that in this case, it is the divergence of the SGS stress tensor, r^,that is modeled [JHF 98] [SCF 98]. For the dynamic models, the model parameter is calculated dynamically at each point from the instantaneous flow field. Negative values of this parameter enable back-scatter, which implies that turbulent energy can be transferred intermittently from the small scales to the larger ones. Local filtering, and artificial bounds, have to be used, in order to avoid large oscillations in the model parameter.

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4. Numerical Algorithm Residual Distribution (RD) or Fluctuation Splitting (FS) schemes date back to the late 80's [ROE 82], when P.L. Roe developed schemes for solving scalar advection-diffusion problems on unstructured meshes. More recently, Van der Weide and H.Deconinck [PDE 97] proposed a matrix generalization of the scalar schemes which can be applied to the solution of non diagonalizable hyperbolic systems. Wood proved [WKK 98] that RD schemes give a better accuracy of the solution, for advection diffusion problems, when compared with the classical finite-volume formulations. Though most applications of RD schemes have been limited so far to steady state computations [DSB 94], we extended their application to accurately simulate time dependent flows [CAO 99] [CAB 99]. Our code uses a second order in time, implicit scheme of Jameson-type [JAM 91]. This is a dual-time step scheme, with sub-iterations to converge the solution at every new real-time step. Multigrid iterations are performed to accelerate the convergence of the solution in these sub-iterations. The code works on an originally developed unstructured mesh, cell-vertex data structure. A hierarchical oct-tree organization of the data was employed with a single tetrahedron division rule. This was shown to allow significant reductions in memory usage, [SCL 97].The discretization of the convective term of the Navier-Stokes equations is based on the matrix Residual Distribution schemes approach [PDE 97]. A finite element, central-Galerkin scheme has been used to discretize the viscous part, the unstationary term and the source terms, i.e. Coriolis forces, centrifugal forces, etc. The update scheme is u™+l

(4)

where U = ( is the conservative variables vector, j = 1..3, AT is the pseudo-time step, n - is the real-time step index, T is one of the cells (tetrahedral) sharing the node i, i — 1..4 is the node index, Vi is the volume of the dual cell i7;, surrounding the i node. Here is the approximation of t/"+1 at the pseudo-time step k, (3f is the modified matrix distribution coefficient for the convective term, iscid advective-residual over the cell T, (5)

where F/ = F?(Un+l>k) is the advective flux vector, and FJ = F?(Un+l>k) is the viscous flux vector, both computed for {Jn+l>k^ nj^ are the cell-face inward normal vectors, ^(^ • id is the contribution of the advective part

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of the Navier-Stokes equations to the nodal residual, ^(FJn^), the contribution of the viscous part of the Navier-Stokes equations to the nodal residual, llli}implicit *s tne volume average of the unstationary term, on the (T n fii). We used an implicit second order discretization with finite differences for [^f]:

A modified second order matrix Residual Distribution scheme has been used for the convective part of the Navier Stokes equations. It employs a dynamic correction of the artificial dissipation of the scheme based on the local flow characteristics. When doing LES, the artificial dissipation of the scheme, i.e. the LW scheme [PDE 97] in our case, is not desired and it can be shut off in regions far from shock waves. Close to shock waves, the artificial dissipation of the original scheme, is reintroduced. If strong shock waves are present, a continuous blend between the second order scheme and a positive first order scheme, i.e. the Narrow scheme, is used in the shock wave region. This enabled us to have monotone shock capturing, while still using very small dissipation in the smooth flow regions. The matrix distribution coefficients, /3/\ are given by:

The matrix distribution coefficients are computed, for the second order scheme, LW:

and for the positive first order, Narrow scheme:

Here vceu is a cell-CFL number. Ki are the Jacobians of the convective dFc flux vector relative to face i, Ki = -Qjj-nj^. The coefficients vcei\, and attend are corrected dynamically. When doing LES, they vanish in regions far from flow field discontinuities. More details about the matrix generalization of the Residual Distribution schemes can be found in [PDE 97]. 5. Parallel Algorithm

Since the Fluctuation Splitting algorithm only needs to access the first order neighbors to update the solution at a vertex, the parallelization process is

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greatly simplified. If it were necessary to access the second-order neighbors, or even higher-order neighbors, then for any vertex that lies along a boundary between processors, a very elaborate and likely expensive method for communicating the appropriate update information across these inter-partition boundaries would need to be devised. Second order classical finite volume codes require access to the second-order neighbors. However, these algorithms get around the complication of having to store the communication information to reach these second order neighbors across processors by using a two-step update procedure. This requires also two communication cycles per time step [VNK 92], whereas the Fluctuation Splitting algorithm only needs one. This makes the fluctuation splitting method very attractive for parallel implementation. The Parallel Virtual Machine (PVM) defacto standard has been used to implement the parallel algorithm in our code [CAD 99]. The code runs as effectively on large distributed memory machines (Cray T3E, IBM SP2...), on the SGI Origin 2000 servers or on a heterogenous collection of platforms. The code takes advantage of its unstructured grid for realizing an almost perfect load balancing by assigning a number of volumes to compute to each processor, proportional with the processors speed. The parallelization of the code has been done using the SPMD paradigm, i.e. the same code is executed in all processors. The global data structure, i.e. grid, field variables, etc. is decomposed into a number of subdomains, equal with the number of running tasks. While any algorithm for domain decomposition can be employed, so far only one dimensional domain decomposition has been used. This is done in a separate preprocessing step and does not affect the solvers flexibility. Each task runs with its own data structure, which is a disjunct part of the global data structure. The tasks are implicitly synchronized by using blocked data exchange. The algorithm implies at each real time step sub-iterations for converging the solution. In each sub-iteration, the solver: - computes the residual, i.e. the integral of the convective, diffusive, source term and unstationary part of the Navier-Stokes equation, over one tetrahedral cell, looping through all cells. Splits the residual according with the distribution scheme in fluctuations, and sends fluctuations to the nodes, where these are collected in some special fields. - communicates using PVM, with neighboring parallel processes, the fluctuations accumulated in nodes and some other useful information, for nodes laying on the domains interface. - imposes boundary conditions, - loops through all nodes, to update the nodal values using the fluctuations stored in those special fields. As it can be seen, the algorithm needs no communication while computing and distributing the fluctuations. This step in the solver algorithm requires the longest computational time. It requires only one back and forward communication for each sub-iteration before applying the boundary conditions and updating the nodal values. The volume size of this communication is reduced, since only nodes on the domains interface exchange information.

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6. Numerical Results LES simulations of complex turbulent flows in engineering applications, using the present code, have been presented in [CAO 99] [CAB 99]. Here we only address the parallel scalability of code and the accuracy of our LES computations.

6.1. Parallel scalability The code has been run on an SGI Origin 2000 platform, on up to 8 processors, and also on a cluster of 2 WinNT PC's, using an Ethernet connection. The results, addressing the parallel speed-up are presented graphically in figure 1. It can be seen that the performance, when using the two WinNT PC cluster, has been close to 75%, due to poor communication while using the Ethernet connection. The code proved instead to have excellent scalability on the Origin platform (98%) [CAD 99].

6.2. LES of channel flow Results for the classical turbulent channel flow are presented. The Reynolds number based on the bulk velocity and the channel height, is 5800. No-slip isothermal boundary conditions have been set at the walls, while periodicity boundary conditions have been used both in the streamwise (x) and the spanwise (z) directions. Computations have been performed using the dynamic DDM model. The grid size is (2-7r<5) (6) (2?r<5/3) where 8 is the channel height. The grid has (85)(51)(51) nodes and aprox. 1,240,000 tetrahedral cells. The grid is stretched in the y—direction (yl = <5(1 — cos(/?;))/2, for pi = 7r(j — l)/(N — 1), j = 1, 2,...., N. Here N is the number of grid points in the y—direction).A uniform grid has been used in the streamwise and the spanwise directions. The flow has been simulated for enough flow-through times (the domain length in the streamwise direction divided by the bulk velocity) to obtain a statistically stationary turbulent channel flow. Data for statistics are accumulated over the last 15 flow-through times. Planar averages are calculated by averaging for all points on planes parallel to the walls and in time, and the results are presented as a function of wall-distance (y) only, and are compared with experiments (Kreplin, 1979) and with DNS data (Kim, 1987). The averaged non-dimensional friction velocity UT = 0.06156 (i.e. uT/Ub, where Ub is the bulk velocity) compares well with the DNS and the experimental (i.e. UT = 0.0643) result. Figures 2 to 5 show planar averaged time averages of the mean of the axial velocity and Root-Mean-Square of velocity fluctuations (RMS of Favre U",V",W"). The results are normalized using the friction velocity, UT. The results obtained using the DDM model compare well with the DNS and the experimental results. Figure 6 shows an instantaneous dis-

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tribution of the skin friction coefficient, C/.

7. Future Work Future work will include extensive testing of different dynamic SGS models. The goal is to perform parallel LES for complex engineering applications.

[PIO 98]

PlOMELLl, U., Large Eddy Simulation: present state and future directions

[ROE 82]

ROE, P.L., Fluctuations and signals. A framework for numerical evolution problems, K.W. Morton and M.J. Baines, editors, Numerical Methods for Fluid Dynamics, Academic Press,

1982. [PDE 97]

PAILLERE, H. et al., Upwind residual distribution methods for compressible flows : An alternative for finite volume and finite element methods, VKI 28th CFD Lecture Series, March 1997.

[DSB 94]

DECONINCK, H. et al., High resolution shock capturing cell vertex advection schemes on unstructured grids, VKI Lecture Series, March 21-25 , 1994.

[WKK 98]

WOOD, WILLIAM A. et al., Diffusion characteristics of upwind schemes on unstructured Triangulations, AIAA 98-2443, Albuquerque, NM 1998.

[JAM 91]

JAMESON, A., Time dependent computations using multigrid, with applications to unsteady flows past airfoils and wings, AIAA paper, 91-1596, 1991.

[CAB 99]

CARAENI, D. et al., LES of spray in compressible flows on Unstructured Grids, AIAA 99-3762, Norfolk, 1999.

[CAO 99]

CARAENI, D. et al., Large Eddy Simulation of the Flow in a Bladed Diffuser,r, TSFP, Santa Barbara, 1999

[CAD 99]

CARAENI, D. et al., Parallel NAS3D: An efficient algorithm for engineering spray simulations using LES, pres. Cl-C, International Parallel CFD'99, 1999, WiUiamsburg.

[SCL 97]

MlTRAN, S. et al., Large Eddy Simulation of rotor stator interaction in centrifugal impeller, JPC, Seattle, July 1997.

[JHF 98]

HELD, J. et al., Large Eddy Simulation of separated transonic flows around a wing section, AIAA 98-0405, Reno 1998.

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[SCF 98]

CONWAY, S. et al., Investigation of the flow across a swirl generator using LES, AIAA 98-0921, Reno 1998.

[LIL 91]

LILLY, D.K., A proposed modification of the Germano subgridscale closure method, Phys. Fluids A, 4:633-635, 1991.

[VNK 92]

VENKATAKRISHNAN V. et al., A MIMD implementation of a parallel Euler solver for unstructured grids, The Journal of Supercomputing, Vol. 6, 1992.

Fig. 1 Parallel SpeedUp

Fig 2. Planar Time Averages of U

Fig 3. Planar averages of U"rms

Fig 4. Planar averages of V'rms

Fig 5. Planar averages of W'rms

Fig. 6 Skin friction distribution

A higher-order-accurate upwind method for 2D compressible flows on cell-vertex unstructured grids

L. A. Catalano Istituto di Macchine ed Energetica Politecnico di Bari Via Re David 200, 70125 Bari, ITALY E-mail: [email protected]

ABSTRACT A finite-volume method for the solution of two-dimensional inviscid compressible flows on cell-vertex unstructured grids is presented. The method is based on a novel bi-linear reconstruction of the unknowns and on a standard flux-difference-splitting scheme. Moreover, a new approach is proposed to achieve the same higher-order accuracy also near solid walls. The method is validated by computing the inviscid flow in a two-dimensional cascade in subsonic and transonic conditions. Key Words: higher-order reconstruction, upwind, unstructured.

1. Introduction In the last decade, a great effort has been devoted by many CFD researchers to the development of higher-order-accurate upwind solvers on unstructured grids. Different approaches have been proposed, including both finite-element and finite-volume discretizations [BAR 91, DEC 92, CAT 97, HAL 97, SEL 96], the major difficulty being the discretization of the inviscid terms in the conservation equations. Concerning the finite volume discretization, most of the upwind schemes proposed to date are based on a gradient-based reconstruction of the flow variables onto the two sides of the surface which defines the finite volume built around each node. Then, an approximate Riemann solver is applied at each interface to select the upwind contributions.

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How to perform a higher-order reconstruction on an unstructured triangular grid is not trivial: if the unknowns are located in the center of each triangle, the finite volume is the cell itself, and the higher-order reconstruction onto its sides becomes a very cumbersome task, because of the difficulty of defining and computing the flow gradients. For such a reason, many researchers have redirect their efforts to cell-vertex discretizations: a dual mesh is first constructed by tracing the medians of each cell, so as to build a finite volume around each node [BAR 93, SEL 96, HAL 97]. Clearly, the interfaces are now located at the midpoint of each side. To knowledge of the author, all of these methods base the reconstruction on a nodal definition of the flow variable gradient. Different approaches have been proposed to compute its value from the gradients in the surrounding cells, but all of them appear complex and time-consuming, see e.g. [BAR 93, HAL 97]. In this paper, an alternative, much simpler approach is proposed, which also includes a new higher-order near-wall discretization. 2. Numerical method 2.1. Governing equations The governing equations for two-dimensional, compressible, inviscid flows, are written in integral form as: (1)

In eq. 1, n is the inward normal of the contour of 5, dS, U = (p, pu, pv, pe°)T is the vector of the conservative variables and F-n = [(pvn), (puvn+pnx), (pvvn + pny), (ph°Vn)]T is the flux entering through the unit length of dS. As usual, p is the density, p is the pressure, e° is the total internal energy and h° is the total enthalpy. Moreover, v will denote the velocity vector, with normal and tangential components vn — v • n and vs = v • s, respectively, and with Cartesian components u and v. The system of governing equations is closed by assuming perfect gas.

2.2. Space discretization The domain is discretized by means of an unstructured mesh composed of triangles with unknows located at each cell-vertex. In particular, the primitive variable vector Q — (p,u,v,p)T is chosen as unknown. A finite volume is constructed around each internal node by connecting the barycenters of two neighbouring triangles, see the node i in fig. 1. An upwind discretization of the RHS of eq. 1 is then obtained as follows: a left state and a right state are reconstructed at the interface (ij] defined on each side connecting the node i

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interface (ij)

auxiliary cells Figure 1: Construction of finite volumes for internal and boundary nodes. Determination of the cell Cji. Construction of the auxiliary cells. with each sorrounding node j. Then, a standard Riemann solver is used to select the upwind contributions to the flux. The left and the right states can be accurately reconstructed by employing a bi-linear variation of Q: (2)

In eq. 2, Iji arid are the two opposite vectors pointing from the two nodes to the intersection of the interface with the side, see fig. 1. Concerning the definition of the gradients, first consider the interface (i + 1/2) of a uniform one-dimensional grid; in such a case, Qf+1/2 is linearly reconstructed as: (3)

It must be remarked that the reconstruction is based on the gradient of Q in the left-neighbouring cell, rather than on the gradient defined in the node i. Similarly, in two dimensions, the gradient (^Q)ji (similar arguments hold for (VQ)jj) must be defined in one of the cells sharing the node j, rather than in the node itself, as it suggested in [BAR 93, HAL 97]. The extension of the previous arguments to two dimensions suggests to define (VQ)ji as the gradient in the cell Cji which contains the prolongation of the side (ji), plotted as a dot-dashed line in fig. 1. Clearly, the choice of a cell-vertex triangular grid allows to define a bi-linear variation of Q in each cell, thus defining the cell gradient (VQ)ji = (VQ}Cji uniquely: (4)

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In eq. 4, Qf. is the primitive variable vector in the node k. Moreover, the opposite side k has length s^ and inward normal n/t. Standard one-dimensional limiters can also be applied to the gradient, but have not been introduced yet. The flux-difference-splitting of Roe [ROE 86] is then used to solve the Riemann problem defined at each interface. The flux is computed as (5)

In eq. 5, 0:^, k = 1, ...,4, are the intensities of the entropy, of the shear and of the two acoustic waves, and A&, k = I,..., 4, are the corresponding propagation velocities:

(6)

Finally, Cfc, k — 1,..., 4 are the eigenvectors which project each wave contribution onto the conservative variable vector:

(7)

In eqq. 6 and 7, <5( ) = ()# — ( )L, and ~ denotes the Roe averages:

(8)

2.2. Boundary conditions When a side lies on a boundary, the construction of the finite volumes is completed by tracing one third of the corresponding median and by adding half of the side to the finite-volume contours of its two nodes: as an example, fig. 1 shows the resulting finite volume associated to the boundary node b. Clearly,

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the flux through each boundary side is computed directly by means of the nodal values; moreover, solid boundaries need only to add the corresponding pressure forces to the momentum equation. Characteristic boundary conditions are then applied to the nodal residuals to ensure that physical boundary conditions are satisfied. The lack of further cells beyond the boundaries make impossible to perform a higher-order reconstruction onto the near-boundary interfaces. The reduced

Figure 2: Computational mesh. accuracy is negligible for far-field boundaries, but a significant amount of numerical entropy would be generated near solid walls. A simple procedure to overcome this problem is here proposed: a row of auxiliary cells is created by adding an isosceles triangle beyond each solid face and by connecting the corresponding auxiliary nodes, see the dashed cells in fig. 1. The states in the auxiliary nodes are updated by imposing the following conditions of Isentropic Simple Radial Equilibrium at the mid-point of the solid face:

(9)

Rc being the radius of curvature at wall. 2.3. Time integration The state in each node is updated by means of a two-stage Runge-Kutta explicit scheme with non-optimal coefficients 0.4 and 1 and CFL number 0.35.

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3. Results The method previously described has been applied to the computation of the inviscid flow in a cascade of VKI LS-59 gas turbine rotor blades, for two different values of the outlet pressure. The blade profile has been modified at the trailing edge by adding an artificial wedge (having no load) in order to simulate the presence of the recirculation zone, which cannot be captured by the present inviscid formulation. Fig. 2 shows the grid used in both the cases analyzed, obtained by slightly modifying a structured grid with 129 x 17 nodes (97 nodes are located on the blade surface). Clearly, the resulting quality of the grid is rather poor. The Mach number contours obtained for outlet Mis

Figure 3: Mach number contours for outlet Mis = 0.81 (AM = 0.05,). (isentropic Mach number) equal to 0.81 are shown in fig. 3. A good agreement

Figure 4: Experimental and computed distributions of Mis on the blade for outlet Mis =0.81. between the computed distribution of Mis on the blade and the experimental one [KIO 86] is visible in fig. 4.

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Figure 5: Mach number contours for outlet M{S — 1 ("AM = 0.05J. The transonic flow with outlet MJS equal to 1 has been then considered: the Mach number contours in fig. 5 show the formation of two shocks on the suction side of the blade, which are both sharply captured. Small oscillations are present and can be eliminated by applying a standard limiter. The surface Mach number distribution shown in fig. 6 is again in good agreement with the experimental data of [KIO 86], except for the discrepancy on the suction side of the blade, due to the presence of a separation bubble caused by the shockboundary layer interaction. Such a phenomenon is obviously missed by the present inviscid analysis.

Figure 6: Experimental and computed distributions of MIS on the blade for outlet Mis = 1-

3. Conclusions A finite-volume method for the solution of two-dimensional inviscid com-

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pressible flows on cell-vertex unstructured grids has been presented. In particular, a novel approach to the higher-order reconstruction of the unknowns has been proposed, and extended to the near-wall regions by means of a row of auxiliary cells. The method has been validated by computing the inviscid flow in a two-dimensional turbine cascade in subsonic and transonic conditions. The extension to the discretization of the viscous-flow conservation equations can be based on state-of-the-art methodologies. 4. Bibliography

[BAR 91]

BARTH T. J., Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations, Lecture Series 1991-06, Von Karman Institute, 1990.

[BAR 93]

BARTH T. J., A 3-D least-squares upwind Euler solver for unstructured meshes, Lecture Notes in Physics, 414, Springer Verlag, pp. 90-94, 1993.

[CAT 97]

CATALANO L. A. et a/., Genuinely multidimensional upwind methods for accurate and efficient solutions of compressible flows, Euler and Navier-Stokes solvers using multidimensional upwind schemes and multigrid acceleration, Notes on Numerical Fluid Mechanics, 57 Vieweg, Braunschweig, Germany,, 1997.

[DEC 92]

DECONINCK H. et a/., Multidimensional upwind methods for unstructured grids, Agard R-787, May, 1992.

[HAL 97]

HALLO L. et a/., An implicit mixed finite-volume-finiteelement method for solving 3D turbulent compressible flows, International Journal for Numerical Methods in Fluids, 25, pp. 1241-1261, 1997.

[KIO 86]

KlOCK R. et a/., The transonic flow through a plane turbine cascade as measured in four European wind tunnels, Transactions of the ASME, Journal of Engineering for gas turbines and power, 108, No. 2, pp. 277-284, 1986.

[ROE 86]

ROE P. L., Characteristic based schemes for the Euler equations, Ann. Rev. Fluid Mech., 18, pp. 337-365, 1986.

[SEL 96]

SELMIN V., FORMAGGIA L., Unified construction of finite element and finite volume discretizations for compressible flows, International Journal for Numerical Methods in Engineering, 39, pp. 1-32, 1996.

A New Upwind Least Squares Finite Difference Scheme (LSFD-U) for Euler Equations of Gas Dynamics N. Balakrishnan

Praveen. C

Assistant Professor Graduate Student CFD Centre, Department of Aerospace Engineering Indian Institute of Science, Bangalore - 560 012 ABSTRACT A new upwind Least Squares Finite Difference Scheme(LSFD-\J) has been developed. The fundamental principle underlying this method is the representation of the derivatives of the fluxes appearing in the conservation laws, using a Generalised Finite Difference strategy based on the method of least squares. This method can operate upon any kind of grid and requires only local connectivity information pertaining to a cloud of grid points around any given node. The results obtained are very encouraging and the use of LSFD-U in the computation of flows past complex configurations is extremely promising. Key Words: Finite difference, Finite volume, Upwind schemes, Hyperbolic equations, Method of Least Squares. 1. Introduction One of the remarkable progresses made in the area of CFD, in recent years is the development of Grid Free Method [1-4] for numerically solving the conservation laws encountered in fluid dynamics. The fundamental principle underlying this method is the representation of the derivatives of the fluxes appearing in the conservation laws using a generalised finite difference strategy based on the method of least squares. This method which can operate upon any kind of grid (structured, unstructured or cartesian) requires only local connectivity information pertaining to a cloud of grid points around any given node. The utility of this method in computing flow past complex configurations is extremely promising. The present work draws its inspiration from the fact that this method has been applied only in the framework of flux vector splitting schemes and not in the framework of flux difference splitting schemes. Here we have attempted to extend the applicability of this Grid Free Method to the framework of flux difference splitting schemes and in the process arrived at an entirely new methodology equally applicable to flux vector splitting schemes. In section 2, we present briefly the details regarding the Least Squares Kinetic Upwind Method (LSKUM) [1-4]. In section 3, the new least squares scheme is presented with a brief review of the flux difference splitting schemes. In section 4, we present the difficulties in extending the methodology to 2D and 3D flows, and also present two variations of the scheme, which would circumvent such problems. In section 5, we present the results and discussions. Concluding remarks are made in section 6. 2. Least Squares Kinetic Upwind Method (LSKUM) The kinetic schemes for solving Euler equations of gas dynamics are obtained by exploiting the fact that these equations are moments of the Boltzmann equation of kinetic theory for gases. Consider the ID Boltzmann equation : f, + vf x = 0 (1) where f is the velocity distribution function which is a Maxwellian and v is the molecular velocity. The fundamental principle underlying LSKUM is that the discrete approximation to f x appearing in the Boltzmann equation is obtained using a least squares approximation given

by,

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(2)

where Afj=fj-f 0 ; AXj=Xj-x0, subscript V denotes the node under consideration and 'j' its neighbour. The moment of the discretized Boltzmann equation will lead to an upwind scheme for the Euler equation, if the stencil of the grid points to be used in equation (2) is chosen, taking into account the direction of signal propagation. In other words, discrete approximation to fx at any given point is obtained by using the grid points on its left if v>0 and vice versa. An interested reader is referred to the papers cited above for a number of interesting developments in LSKUM, such as the use of weighted least squares in equation (2). This idea when extended to the flux vector split framework of Euler equations, given by, Ut+Fx++Fx- = 0 (3) where U is the vector of conserved variables and F is the flux vector, the discrete approximation to F x *at any given point will involve grid points to its left, and Fx~ will involve grid points to its right. 3. Upwind Least Squares Finite Difference Method (LSFD-U) Inspired by the fact that the discrete least squares approximation to the derivative Fx involves the flux difference term AF, it was thought that it would be appropriate to make use of flux difference splitting in the least squares framework. Before we discuss the details of the present least squares algorithm, we briefly discuss the flux difference splitting scheme as applied to finite volume framework.

Fig 1.

Typical ID Finite Volume Computational Domain

Fig 1.depicts a typical ID finite volume computational domain. In the flux difference splitting scheme[5], the total flux difference AF = FR - FL is split into a positive part (AF)+ corresponding to the right running waves and a negative part (AF)" corresponding to the left running waves, based on a suitable linearization procedure, in such a way that the interfacial flux F( is given by (4)

In a finite volume framework, an interfacial flux obtained as an average of the above two expressions is made use of in the state update formula. Now we describe the present methodology. Equation 4 given above clearly suggests that the flux difference between any fictitious interface perpendicularly intersecting the line connecting the two grid points and the points themselves, can be obtained using a suitable linearization procedure. At the heart of the present methodology is the use of such flux differences based on upwinding principle in the discrete least squares approximation to Fx appearing in the Euler equations. This leads to an upwind scheme based on the least squares principle. Also, it is not necessary that the present methodology should be used only in conjunction with Flux Difference Splitting Schemes. The very fact that an upwind estimate of

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the flux at the fictitious interface is all that is required for the determination of flux differences involved in the present method, suggests the use of this method in conjunction with all upwind flux formula. As can be easily seen, the present methodology makes use of a global stencil of grid points in contrast to the methodology described in section 2, which requires an upwind stencil of grid points. Also, we make use of an upwind estimate of the interfacial flux for determining AF's appearing in the least squares formula, in contrast to the explicit use of the nodal values of the fluxes in the earlier framework. 4. LSFD-U in 2D Here we present some of the interesting problems we face when extending the algorithm just described for solving 2D flows. Fig.2 gives a typical 2D stencil of grid points. Let Ij represent the fictitious interface drawn across the line connecting the point under consideration 'o', and its jth neighbour, and n : represent the unit vector along

'oj'.

Representing the flux along oj by

we have (5) (6)

where F and G are fluxes in x and y directions respectively. Now our job reduces to recovering the information regarding the gradients of the 2D fluxes namely VF0 and VG fr°m tne many ID flux difference terms given in equation 6. The derivatives Fx and Gy thus recovered would eventually be used in the state update formula. Equation 6 represents an over determined system of equations and it appears that the straight forward way to obtain the gradients of the fluxes is to minimise X E • w^h respect to the derivatives of the fluxes, where, (7)

Unfortunately, simple algebra would demonstrate that such a procedure leads to a singular system. To circumvent this problem we suggest the following two methodologies. 4.1 Method 1 Method 1 draws its inspiration from the work of Ghosh and Deshpande [2]. Here we effect locally a co-ordinate transformation (x,y)->(^,T|), in such a way that one of the axes (Q coincides with the streamwise direction. It is a well known fact that the fluxes normal to the streamwise co-ordinate direction involve only pressure terms and a global stencil of grid points can be used for approximating the derivatives of such fluxes without loss in stability. If F and G represent the fluxes along the new co-ordinate directions E, and r\ respectively, the derivatives Gc and G can be obtained using the least squares procedure described in [3], S T\ making use of a global stencil of grid points. Thus the streamwise rotation of the co-ordinates leaves us with a non singular system involving Ft and Fn • A simple least squares procedure described in the previous section can be adopted for the estimation of R and R,. The derivatives R and G thus determined are substituted in the state update formula. It should

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be remarked that in this method, the second and third components of the fluxes F and G represent ^ and T] momentum conservation. 4.2 Method 2 In this method we locally rotate the co-ordinate system from (x,y) —^ (^,r|),

m

such a way that p + G& = 0- where F and G represent the fluxes along the new coordinate directions £, and r\ respectively. Note that the second and third components of the flux vectors Fand G still represent the x and y momentum conservation. It can easily be demonstrated that a co-ordinate system rotated at an angle p\ given by, (8)

would satisfy the condition that p + Gt = 0 • Th£ gradients of the fluxes used in the estimation of (5 are obtained using the least squares procedure [3] making use of a global stencil of grid points. This results in an overdetermined system of equations with two unknowns Ft and G~ , which can be solved using least squares procedure. The derivatives thus determined are used in the state update formula. 5. Results and Discussions The new least squares upwind finite difference method (LSFD-U) is validated using standard ID and 2D test problems. In the computations high resolution is obtained based on a linear reconstruction procedure [6]. Non physical oscillations in the solution are suppressed using Yenkatakrishnan limiter [7]. In all the computations presented in this work the fictitious interface is always placed at the mid point of the line segment under consideration. Figure 3 gives the results obtained for the ID shock tube problem of Sod [8]. The results are obtained on a non-uniform grid generated using cosine spacing for grid points. One hundred grid points have been used in the computation. Roe [5] flux has been used in these calculations. The grids made use of in the 2D computations are presented in Figure 4. The grid details are given in Table 1. All the 2D computations have been made with KFVS [9] flux formula.

Configuration Cylinder NACA0012

Table 1 No. of nodes 4317 4733

No. of nodes on the wall 160 160

The pressure contours obtained for a low subsonic flow past a cylinder(in the incompressible limit; M«=0.1) are presented in Figures 5 and 6. The wall pressure data are compared with the exact potential flow solution in Figure 7. The Mach contours obtained for subsonic(Mro=0.63, angle of attack=2°) and transonic (MM=0.85, angle of attack=l°) flows past NACA0012 airfoil are presented in Figures 8,9,11 and 12. The CP values are compared with those obtained using the cell vertex finite volume scheme in Figures 10 and 13. The first order results obtained using Method 2 are identical to those obtained using Method 1, and therefore are not presented here. The CL and CD values obtained using LSFD-U are compared with GAMM[10] results in Table 2.

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Table 2 M.=0.63, AOA=2°

Method

I II GAMM

335

M«=0.85, AOA=1°

CL

CD

CL

CD

0.3387 0.3343 0.3335

0.0014 0.0006 0.0000

0.3772 0.3806 0.3790

0.0550 0.0545 0.0576

From the results it is evident that the new LSFD-U framework is capable of capturing all features expected out of inviscid compressible flows. Also, it should be remarked that the linear reconstruction procedure developed for the finite volume schemes, when used in conjunction with the present LSFD-U framework, can produce excellent improvement in solution resolution. 6. Conclusions A new upwind least squares finite difference method has been developed. The new scheme by the virtue of using a least squares framework can be considered as a Grid Free Method. It has an added advantage of making use of a global stencil. The way the interfacial fluxes are calculated in the new scheme resembles that of finite volume method and therefore all the developments that have taken place in finite volume method, like the method of reconstruction [6] can be directly adopted. The use of LSFD-U in the computation of flows past complex configurations is extremely promising.

Acknowledgements The authors like to thank Mr J. C. Mathur. NAL, Bangalore, for providing the unstructured mesh generator. The authors like to thank Mr. Krishnakumar, under graduate student, 1IT, Kharagpur, for his involvement in the initial phases of development of the code used in the computations, during his stay at IISc, Bangalore, as a Summer trainee. The authors also like to express their gratitude to Mr. Harish. R, project assistant, CFD Centre, IISc, for his invaluable help in preparing the manuscript. References S.M. Deshpande, A.K. Ghosh and J.C Mandal, "Least Squares Weak Upwind Method for Euler Equations", 89 FM 4, Fluid Mechanics Report, Department of Aerospace Engineering, Indian Institute of Science, Bangalore. 2. A.K. Ghosh, "Robust Least Squares Kinetic Upwind Method for Inviscid Compressible Flows", June 1996, PhD thesis. 3. S.M. Deshpande, P.S. Kulkarni, and A.K. Ghosh, "New Developments in Kinetic Schemes", Computers Math. Applic., Vol 35, No 1/2, pp. 75-93, 1998 4. K. Anandhanarayanan, D.B. Dhokrikar, V. Ramesh and S.M. Deshpande, "A Grid Free Method for 2D Euler Computations using Least Squares Kinetic Upwind Method", Proceedings of the third Asian Computational Fluid Dynamics Conference, pp. 390, vol 2, December?"1 - 1 Ilh 1998, Bangalore, India. 5. Roe, Philip. L. "Approximate Reimann Solvers Parameter Vectors and Difference Schemes", Journal of Computational Physics, Vol. 43, pp. 357-372, 1981 6. T.J. Barth " Higher Order Solution of the Euler Equations on Unstructured Grids using Quadratic Reconstruction", AIAA-90-0013, 1990 7. V. Venkatakrishnan, "Convergence of Steady State Solutions of Euler Equations on Unstructured Grids with Limiters", JCP, vol 118, pp. 120, 1995. 8. G.A. Sod, "A Survey of Several Finite Difference Methods for system of Nonlinear Hyperbolic Conservation Laws", JCP, vol 27, pp. 1, 1978. 9. J.C. Mandal and S.M. Deshpande, " Kinetic Flux Vector Splitting for Euler Equations", Computers and Fluids, vol 23, No 2, pp 447-478. 10. GAMM Workshop on Numerical solutions of Compressible Euler Flows, June 1986. 1.

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Fig 2. Typical 2D grid distribution for LSFD-U

Fig 3. LSFD-U applied to shock tube problem

Fig 4. Unstructured grid used for 2D computations Linear Reconstruction

Fig 5 Pressure contours obtained using Method 1 for M»=0.1

Linear Reconstruction

Fig 6 Pressure contours obtained using Method 2 for M>=0.1

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Fig 7 Pressure distribution over a cylinder at M«,=0.1

Fig 8 Mach contours using Method 1; M«=0.63 and AOA=2°

Fig 9 Mach contours using Method 2 for M^O.63 and AOA=2°

Fig 10 Pressure distribution for M«=0.63 and AOA=2°

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Fig 11 Mach contours using Method I; MM=0.85 and AOA=1°

Fig 12 Mach contours using Method 2 for M^O.85 and AOA=1°

Fig 13 Pressure distribution for M»=0.85 and AOA=1°

A finite-volume algorithm for all speed flows F. Moukalled and M. Darwish American University of Beirut, Faculty of Engineering & Architecture, Mechanical Engineering Department, P.O.Box 11-0236, Beirut, Lebanon. ABSTRACT. A new collocated finite volume-based solution procedure for predicting viscous compressible and incompressible flows is presented. The technique is equally applicable in the subsonic, transonic, and supersonic regimes. Pressure is selected as a dependent variable in preference to density because changes in pressure are significant at all speeds as opposed to variations in density which become very small at low Mach numbers. The newly developed algorithm has two new features; (i) the use of the Normalized Variable and Space Formulation methodology to bound the convective fluxes; and (ii) the use of a HighResolution scheme in calculating interface density values to enhance the shock capturing property of the algorithm. Keywords: Pressure-based method, All speed flows, High-Resolution algorithm.

1. Introduction In Computational Fluid Dynamics (CFD) a great research effort has been devoted to the development of accurate and efficient numerical algorithms suitable for solving flows in the various Reynolds and Mach number regimes. The type of convection scheme to be used in a given application depends on the value of Reynolds number. On the other hand, the Mach number value dictates the type of algorithm to be utilized in the solution procedure. These algorithms can be classified into two groups: density-based methods and pressure-based methods, with the former used for high Mach number flows, and the latter for low Mach number flows. The ultimate goal however, is to develop a unified algorithm capable of solving flow problems in the various Reynolds and Mach number regimes. To understand the difficulty associated with the design of such an algorithm, it is important to understand the role of pressure in compressible flow [KAR 86]. In the low Mach number limit where density becomes constant, the role of pressure is to act on velocity through continuity so that conservation of mass is satisfied. Obviously, for low speed flows, the pressure gradient needed to drive the velocities through momentum conservation is of such magnitude that the density is not affected significantly and the flow can be considered nearly incompressible. Hence, density and pressure are very weakly related. As a result, the continuity equation is decoupled from the momentum equations and can no longer be considered as the equation for density. Rather, it acts as a constraint on the velocity field. Thus, for a sequential solution of the equations, it is necessary to devise a mechanism to couple the continuity and momentum equations through the pressure field. In the

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hypersonic limit where variations in velocity become relatively small as compared to the velocity itself, the changes in pressure do significantly affect density. In this limit, the pressure can be viewed to act on density alone through the equation of state so that mass conservation is satisfied [K.AR 86] and the continuity equation can be viewed as the equation for density. The above discussion reveals that for any numerical method to be capable of predicting both incompressible and compressible fluid flows the pressure should always be allowed to play its dual role and to act on both velocity and density to satisfy continuity. Several researchers [KAR 86, RHI 86, MAR 94, DEM 93, LIE 93] have worked on extending the range of pressure-based methods to high Mach numbers. In most of the published work the first order upwind scheme is used to interpolate for density, exception being in the work presented in [DEM 93] where a central difference scheme blended with the upwind scheme is used. The bleeding relies on a factor varying between 0 and 1, which is problem dependent and has to be adjusted to eliminate oscillation or to promote convergence. In the work presented in [LIE 93] the retarded density concept is utilized in calculating the density at the control volume faces. This concept is based on factors that are also problem dependent and requires the addition of some artificial dissipation to stabilize the algorithm (second-order terms were introduced), which complicate its use. To this end, the objective of this paper is to present a newly developed collocated pressure-based solution procedure that is equally valid at all Reynolds and Mach number values. The algorithm will have two new features. The first one is the use of the Normalized Variable Formulation (NVF) [LEO 87] and/or the Normalized Variable and Space Formulation (NVSF) [DAR 94] methodology in the discretization of the convective terms. The second one, is the use of HighResolution (HR) schemes in the interpolation of density in the source of the pressure correction equation and the convective fluxes in order to enhance the shock capturing capability of the method. The increase in accuracy with the use of HR schemes for density is demonstrated by comparing predictions, for the flow over a bump, obtained using the third-order SMART scheme for all variables except density (for which the Upwind scheme is used) against another set of results obtained using the SMART scheme for all variables including density. 2. Finite volume discretization of the transport equations The conservation equations governing two-dimensional compressible flow problems may be expressed in the following general form:

where (j) is any dependent variable, v is the velocity vector, and p, F*, and Q* are the density (=P/RT), diffusivity, and source terms, respectively. Integrating the above equation over a control volume (Fig. 1) and applying the divergence theorem, the following discretized equation is obtained:

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where J f represents the total flux of <j> across face T and is given by Each of the surface fluxes Jf contains a convective contribution, JJ? , and a diffusive contribution, jj?, hence: where The diffusive flux at the control volume face 'f is discretized using a linear symmetric interpolation profile so as to write the gradient as a function of the neighboring grid points. The convective flux across face f can be written as: where Cf is the convective flux coefficient at cell face T. As can be seen from [6] the accuracy of the control volume solution for the convective scalar flux depends on the proper estimation of the face value <j)f as a function of the neighboring § nodes values. Using some assumed interpolation profile, (j>f can be explicitly formulated in terms of its node values by a functional relationship of the form: where <|)nb denotes the neighboring node (j> values. After substituting [7] into [6] for each cell face and using the resulting equation along with the discretized form of the diffusive flux, [2] is transformed after some algebraic manipulations into the following discretized equation:

where the coefficients ap and a NB depend on the selected scheme and bp is the source term of the discretized equation. \

V

I

Figure 1. Control volume. 3. The NVSF methodology for constructing HR schemes As mentioned earlier, the discretization of the convection flux is not straightforward and requires additional attention. Since the intention is to develop a high-resolution algorithm, the highly diffusive first order UPWIND scheme [PAT 81] is excluded. As such, a high order interpolation profile is sought. The difficulties associated with the use of such profiles stem from the conflicting requirements of accuracy, stability, and boundedness. Solutions predicted with high order profiles tend to provoke oscillations in the solution. To suppress these

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oscillations, the composite flux limiter method [LEO 88] is adopted here. The formulation of high-resolution flux limiter schemes on uniform grid has recently been generalized in [LEO 88] through the Normalized Variable Formulation (NVF) methodology and on non-uniform grid in [DAR 94] through the Normalized Variable and Space Formulation (NVSF) methodology. To the authors' knowledge, the NVSF formulation has never been used to bound the convection flux in compressible flows. It is an objective of this work to extend the applicability of this technique to compressible flows. For more details the reader is referred to [DAR 94]. 4. High resolution algorithm The need for a solution algorithm arises in the simulation of flow problems because a scalar equation does not exist for pressure. Hence, if a segregated approach is to be adopted, coupling between the u, v, p, and P primitive variables in the continuity and momentum equations will be required. The segregated algorithm adopted in this work is the SIMPLE algorithm [PAT 81], which involves a predictor and a corrector step. In the predictor step, the velocity field is calculated based on a guessed or estimated pressure field. In the corrector step, a pressure (or a pressure-correction) equation is derived and solved. Then, the variation in the pressure field is accounted for within the momentum equations by corrections to the velocity and density fields. Thus, the velocity, density, and pressure fields are driven, iteratively, to better satisfying the momentum and continuity equations simultaneously and convergence is achieved by repeatedly applying the above-described procedure. The key step in deriving the pressure-correction equation is to notice that in the predictor stage a guessed or estimated pressure field from the previous iteration, denoted by P , is substituted into the momentum equations, the resulting velocity field, denoted by v , which now satisfies the momentum equations, will, in general, not satisfy the continuity equation. Thus, a correction is needed in order to obtain a velocity and pressure fields that satisfy both equations. Denoting the pressure, velocity, and density corrections by P', v'(u', v'), and p', respectively, the corrected fields are given by: Combining momentum and continuity and substituting P, v, and p using [9], the final form of the pressure-correction equation is: where

From [11] it is clear that the starred continuity equation appears as a source term in the pressure correction equation. Moreover, in a pressure-based algorithm, the pressure-correction equation is the most important equation that gives the pressure, upon which all other variables are dependent. Therefore, the solution accuracy depends on the proper estimation of pressure from this equation. Definitely, the

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more accurate the interpolated starred density (p ) values at the control volume faces are, the more accurate the predicted pressure values will be. The use of a central difference scheme for the interpolation of p leads to instability at Mach numbers near or above 1 [KAR 86, DEM 93]. On the other hand the use of a first order upwind scheme lead to excess diffusion [KAR 86]. The obvious solution would be to interpolate for values of p at the control volume faces the same way interpolation for other dependent variables is carried. That is to employ the bounded HR family of schemes for which no problem-dependent factors are required. Adopting this strategy, the discretized form of the starred continuity equation becomes:

The same procedure is also adopted for calculating the density when computing the mass flow rate at a control volume face in the general conservation equation. 5. Results and discussion The validity of the above described solution procedure is demonstrated in this section by presenting solutions to the inviscid flow over a bump. The physical situation consists of a channel of width equal to the length of the circular arc bump and of total length equal to three lengths of the bump. Results are presented for three different types of flow (subsonic, transonic, and supersonic). For subsonic and transonic calculations, the thickness-to-chord ratio is 10% and for supersonic flow calculations it is 4%. In all flow regimes, predictions obtained over a relatively coarse grid using the SMART scheme for all variables including density are compared against results obtained over the same grid using the SMART scheme for all variables except density, for which the UPWIND scheme is used. Due to the unavailability of an exact solution to the problem, a solution using a dense grid is generated and treated as the most accurate solution against which coarse grid results are compared. 5.1 Subsonic flow over a circular arc bump With an inlet Mach number of 0.5, the inviscid flow in the channel is fully subsonic and symmetric across the middle of the bump. Isobars displayed in Fig. 2(a) reveal that the coarse grid solution obtained with the SMART scheme for all variables falls on top of the dense grid solution. The use of the upwind scheme for density however, lowers the overall solution accuracy. The same conclusion can be drawn when comparing the Mach number distribution along the lower and upper walls of the channel. As seen in Fig. 2(b), the coarse grid profile obtained using the SMART scheme for density is closer to the dense grid profile than the one predicted employing the upwind scheme for density. The difference in results between the coarse grid solutions is not large for this test case. This is expected since the flow is subsonic and variations in density are relatively small. Larger differences are anticipated in the transonic and supersonic regimes.

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\~s

Figure 2. Subsonic flow over a 10% circular bump; (a) isobars and (b) profiles along the walls. 5.2 Transonic flow over a circular arc bump Results for an inlet Mach number of 0.675 are displayed in Fig. 3. Figure 3(a) presents a comparison between the coarse grid and dense grid results. As shown, the use of the HR SMART scheme for density greatly improves the predictions. Isobars generated over a coarse grid (63x16 c.v.) using the SMART scheme for all variables are very close to the ones obtained with a dense grid (252x54 c.v.). This is in difference with coarse grid results obtained using the upwind scheme for density and the SMART scheme for all other variables, which noticeably deviate from the dense grid solution. This is further apparent in Fig. 3(b) where Mach number profiles along the lower and upper walls are compared. As shown, the most accurate coarse grid results are those obtained with the SMART scheme for all variables and the worst ones are achieved with the upwind scheme for all variables. The maximum Mach number along the lower wall (si.41), predicted with a dense grid, is in excellent agreement with published values [DEM 93]. By comparing course grid profiles along the lower wall, the all-SMART solution is about 11% more accurate than the solution obtained using SMART for all variables and upwind for density and 21% more accurate than the highly diffusive all-upwind solution. 5.3 Supersonic flow over a circular arc bump Computations are presented for an inlet Mach number value of 1.4. Mach number contours are compared in Fig. 4(a). As before, the course grid all-SMART results (58x18 c.v.), being closer to the dense grid results (158x78 c.v.), are more accurate than those obtained when using the upwind scheme for density. The Mach profiles along the lower and upper walls, depicted in Fig. 4(b), are in excellent

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agreement with published results [NI 82] and reveal good enhancement in accuracy when using the SMART scheme for evaluating interface density values. The use of the upwind scheme to compute density deteriorates the solution accuracy even though a HR scheme is used for other variables. The all-upwind results are highly diffusive.

Figure 3. Transonic flow over a 10% circular bump; (a) isobars using various schemes, and (b) profiles along the walls.

Figure 4. Supersonic /low over a 4% circular bump (Min-1.4); (a) Mach number contours using various schemes, (b) profiles along the walls.

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6. Concluding Remarks A new high-resolution pressure-based algorithm for the solution of fluid flow at all speeds was formulated. The new features in the algorithm are the use of a HR scheme in calculating the density values at the control volume faces and the use of the NVSF methodology for bounding the convection fluxes. Results obtained were very promising and revealed good enhancement in accuracy at high Mach number values when calculating interface density values using a High-Resolution scheme. 7. Acknowlegments The financial support provided by the European Office of Aerospace Research and Development (BOARD) (SPC-99-4003) is gratefully acknowledged. 8. Bibliography [DAR 94] Darwish, M.S. and Moukalled, F.," Normalized Variable and Space Formulation Methodology For High-Resolution Schemes," Numerical Heat Transfer, Part B, vol. 26, pp. 79-96, 1994. [DEM 93] Demirdzic, I., Lilek, Z., and Peric, M.,"A Collocated Finite Volume Method For Predicting Flows at All Speeds," International Journal for Numerical Methods in Fluids, vol. 16, pp. 1029-1050, 1993. [KAR 86] Karki, K.C.,"A Calculation Procedure for Viscous Flows at All Speeds in Complex Geometries," Ph.D. Thesis, University of Minnesota, June 1986. [LEO 87] Leonard, B.P./'Locally Modified Quick Scheme for Highly Convective 2-D and 3-D Flows," Taylor, C. and Morgan, K. (eds.), Numerical Methods in Laminar and Turbulent Flows, Pineridge Press, Swansea, U.K., vol. 15, pp. 35-47, 1987. [LEO 88] Leonard, B.P.,"Simple High-Accuracy Resolution Program for Convective Modelling of Discontinuities," International Journal for Numerical Methods in Engineering, vol. 8, pp. 1291-1318, 1988. [LIE 93] Lien, F.S. and Leschziner, M.A.,"A Pressure-Velocity Solution Strategy for Compressible Flow and Its Application to Shock/Boundary-Layer Interaction Using Second-Moment Turbulence Closure," Journal of Fluids Engineering, vol. 115, pp. 717-725, 1993. [MAR 94] Marchi, C.H. and Maliska, C.R.,"A Non-orthogonal Finite-Volume Methods for the Solution of All Speed Flows Using Co-Located Variables," Numerical Heat Transfer, Part B, vol. 26, pp. 293-311, 1994. [NI 82] Ni, R.H.,"A Multiple Grid Scheme for Solving the Euler Equation," AIAA Journal, vol. 20, pp. 1565-1571, 1982. [PAT 81] Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere, N.Y., 1981. [RHI86] Rhie, C.M.,"A Pressure Based Navier-Stokes Solver Using the Multigrid Method," AIAA paper 86-0207, 1986.

Preserving Vorticity in Finite-Volume Schemes

Philip Roe and Bill Morton Department of Aerospace Engineering The University of Michigan Ann Arbor, Michigan 48109-2140 USA Department of Mathematical Sciences University of Bath Bath BA2 7AY United Kingdom

ABSTRACT We discuss the fact that many otherwise accurate finite-volume schemes have a tendency to yield anomalous solutions in certain circumstances. These are strongly linked to the appearance of spurious vorticity. For a model problem, we show that certain finite volume methods in fact preserve vorticity. Although these are not new schemes, they are not currently fashionable. A possibility exists to modify them so that they are are of high-resolution and upwind with respect to acoustic waves. Key Words: Conservation laws, Euler equations, vorticity

1.

Introduction

Vorticity is a very important aspect of many fluid flows, especially in three dimensions, on account of the great times for which it persists in a highReynolds number flow following its initial creation. The velocities "induced" by vorticity are essential to the operation of many fluid devices,and the behaviour of those regions where vorticity is concentrated are the key to the generation of sound. For computational purposes, incompressible flows are often formulated in variables that include vorticity, and "vortex methods" that explicitly track

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concentrated vortex cores are also widely used. For compressible flow, vorticity is hardly less important, but it is rare for it to be included in the computational formulation. Instead, the stress is placed on conservation so as to ensure the correct capturing of shocks. The consequence is that vortical structures are often badly diffused by numerical dissipation and this is a serious impediment to the prediction of complcated viscous flows. We have recently begun [MOR 99] construction of finite-volume methods that exert control over vorticity. We make no attempt to solve the vorticity evolution equations themselves, but try to find schemes such that a correct evolution of the discrete vorticity is an exact consequence of the discrete conservation laws, just as at the continuum level. Another way of expressing this is to say that whatever form of numerical dissipation is employed to stabilise the propagation of waves should have no impact on the advection of vorticity. There is a further motivation, which is that at the level of the Euler equations many otherwise accurate methods, especially those that attempt to minimise numerical dissipation by adopting upwind strategies (so that any disturbance is damped by the smallest damped compatible with its propagation speed) notoriously produce nonphysical solutions (carbuncle phenomenon) under some special (but not uncommon) circumstances. It now appears that these solutions are in fact quite legitimate solutions of the Euler equations, and could only be excluded by some selection principle over and above that of entropy-satisfaction. On the basis of examples encountered to date, all of these undesirable solutions appear to feature much greater vorticity that the anticipated solution. A very simple example is given in Section 2. There is also the related phenomenon discovered by Quirk [QUI 94]; that the same schemes that give rise to carbuncles can be "destabilsed" by very small grid perturbations. In simple academic examples it is usually easy to detect that something is wrong with the solution, and to tinker with the code until it goes away, but in situations where the flow is anyway expected to be complex there may be no simple way to detect unwanted phenomena. There is therefore an urgent need to understand these anomalies and to find means of avoiding them. 2.

Nonunique Solutions of the Euler Equations

Consider the supersonic flow past a flat-nosed two-dimensional body (although the argument works just as well for a flat-nosed cylinder with axial symmetry). We expect the solution to feature a rather smooth detached bow shock, giving rise to an embedded subsonic region, from which the flow accellerates smoothly to regain supersonic speed. This is shown in the top half of Figure 1. In the bottom half is an alternative solution. Plane oblique shocks originate from a more-or-less arbitrary point on the line of symmetry. Behind them is a triangular region of stagnant fluid extending to the shoulder of the body. Outside of this region the flow behaves exactly as it would if flowing past a triangular wedge (or cone) of solid matter. The pressure, velocity, etc.

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Figure 1. Two possible Euler solutions for the the flow past a flat-faced body. in that region may be found from the usual compressible tables and formulae. Prandtl-Meyer rarefaction waves return the flow to its original direction and weaken the initial shocks. Such a flow is in fact possible to create experimentally, and is illustrated in Plate 272 of [VAN 82] which is a schlieren picture taken in a windtunnel. The unusual flow configuration is the result of a thin "splitter plate" placed along the axis AB. Presumably what happens is that the boundary layer on the plate creates vorticity that is then diffused outward. In the limit of very high Reynolds number the vorticity eventually becomes concentrated in the infinitesimal shear layers AC, AD There is no net circulation generated in the flow but each half of the stagnant region (above and below the axis) has circulation around it. This flow is a valid second solution of the Euler equations, provide that the pressure in the stagnant region matches the pressure outside it. The temperature in the stagnant region is arbitrary if we allow that an entropy layer might also exist, so there is a doubly-infinite family of solutions. All the shocks are entropy-satisfying, but under some conditions the shear layers might not be stable. One may now imagine that the corners of the body are steadily rounded off until the nose becomes a semi-circle. The alternative solutions would become the carbuncles. According to Pandolfi and D'Ambrosio, who have made very detailed observations [PAN 99] the solutions produced in the carbuncle phenomenon are also genuine Euler solutions. One might hope that by solving the Navier-Stokes equations instead the carbuncle would automatically disappear. However, Gressier and Moschetta

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[GRE 99] report that the Reynolds number had to be reduced to as little as 100 before this happened. It seems that the mechanisms by which the Euler code introduces the vorticity is stronger than the mechanism by which the Navier-Stokes code removes it. The Quirk phenomenon occurs when a plane one-dimensional shock is propagated along one direction of a rectangular grid. If the grid is indeed perfectly rectangular then one recovers the one-dimensional solution but if one row of grid points, parallel to the direction of motion, is very slightly perturbed, an exponentially-growing pattern of high-frequency disturbances is generated that eventually yields a flow pattern not unlike that seen in carbuncles. Robinet et al [ROB 99] attempt to relate this to a new observation concerning the linear stabilty of a plane shock. Although this is a classical flow stability problem, they have discovered a 'strange mode' overlooked by previous analysts. They find perturbations of a plane two-dimensional flow by separation of variables. This leads to an eigenvalue problem having linearly independent solutions except for certain combinations of frequency in time and wavenumber in space. Rather unexpectedly, these exceptional solutions involve a resonance in which an acoustic mode and a vortical mode become indistinguishable. The linear algebra problem has to be completed by a Jordan block with a generalised eigenvector. Coupling this with the equations governing the shock perturbation reveals a solution that can grow exponentially with time. Many features of this solution are shown to appear in numerical experiments on the Quirk phenomenon. Experience to date is that all numerical schemes that display the carbuncle phenomenon also display the Quirk phenomenon. There is a proof [GRE 98] that all schemes capable of resolving a parallel shear layer without dissipation (a highly desirable property, and one responsible for the widespread adoption of flux-difference-splitting schemes rather than flux-splitting schemes) will in fact display the Quirk phenomenon. This collection of facts almost suggests a crisis situation. In the next section we present analysis, condensed from [MOR 99] that opens an avenue of escape. 3.

A Model Problem

The simplest model problem to combine wave propagation and vorticity is the system wave equation. We will write this in two space dimensions in the matrix form, using a notation corresponding to acoustic waves in a fluid that is stationary in the mean, with pressure p* and velocity u* = (u*,v*), thus dtu+cLu = Q.

(1)

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Figure 2. Grid definition. Here u = (p*/(pc2), u*/c, v*/c) = ( p , u , v , ) , c is the sound speed in the mean flow, and

Restriction to two dimensions is merely for economy of notation; all of the analysis extends very straightforwardly to three dimensions. We study the wave equation in system rather than scalar (dttu — c 2 V 2 u) form for two reasons. Firstly because this is the form of the wave equation that is hidden inside the Euler equations, whether in their two-dimensional unsteady or in their three-dimensional supersonic steady forms. Secondly, because the scalar form automatically implies vanishing vorticity, whereas the interaction of the waves with vorticity is one of the aspects we want to study. Here the interaction is very simple, as befits a model problem. We easily deduce from (1,2) that dt£ = 0,

where

£ — dxv — dyu.

In other words, there is no interaction and any initial distribution of vorticity is preserved. Maintaining this independence at the discrete level will be our objective. 3.1.

Discrete Notation

In this paper we concentrate on simple finite-difference formulations on uniform square grids, such that the spacing in the x and y directions is h and the time step is At, with ufj a discrete approximation located at ( x , y , t ) —

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(ih,jh,nAt). The standard discrete differencing and averaging operators are defined by

The points with integer coordinates will be called cells, those with one integer and one half-integer coordinate edges and those with two half-integer coordinates vertices. The variables stored at cells will be (p,«,?;). The same variables stored at vertices will be distinguished by primes where neccessary. The variables stored at vertical edges will be (P, U) and those stored at horizontal edges (Q, V). In a finite-volume interpretation the edge quantities are the fluxes. (See Figure 2) If it is recognised that each application of the above operators moves the mesh values to a different set of grid points then arbitrarily long products of operators are allowed and all multiplications commute. 3.2.

Conservation Form

In the cell-centred finite-volume method, discrete conservation is ensured by drawing a control volume around the grid point of interest, and writing the update as an integral around this volume. In the generic case of a vector U of conserved variables, with fluxes F, G in the (x, y)-directions respectively, one has

where F*, G* are numerical fluxes evaluated from some formula to be determined. In the present case we can write, with v = c&i/h and following the notation of Figure 2,

It will usefully restrict the schemes to require that they can be written in this form. A second-order scheme of the Lax-Wendroff type follows from taking U, V, P, Q to be estimates halfway through the time step. However, we will find subsequently that it is a rather special type of conservation form that emerges from the analysis.

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3.3.

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Preserving Vorticity

The first step in creating a scheme that preserves vorticity is to define a discrete vorticity, which we do as follows

This will be preserved if

and the condition ^ixP = ^yQ will be met if we take P = nyr' ,Q = nxr' where r' is some quantity defined at vertices. The only way to define a consistent local pressure while retaining a nine-point stencil is to take

In that case we have

To obtain second-order accuracy, r' must now be updated to half-way through the time step. The simple formula

is the unique symmetrical formula to achieve this without enlarging the stencil, leading to

We remark here that a general example of the Lax- Wendroff family will generate vorticity at a rate proportional to h3; this is shown in [MOR 99].

3.4-

Complete Evolution Operator

We construct the matrix operator that will update the solution, so that if

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certain elements of the matrix MA are already uniquely determined by insisting that the velocities are updated with second-order accuracy while preserving the discrete vorticity (6). We have in fact, from (4),(5),(9),(10),

The adjoint property div/j = —grad£ requires that this matrix be symmetric [MOR 99] and an agument based on maintaining a compact stencil leads to the complete matrix operator

The scheme represented by this matrix has been uniquely determined by the requirements of conservation, vorticity preservation, symmetry of the solution under grid transformations, adjoint symmetry of the discrete operator, and second-order accuracy. However it is not a new scheme. It can be recognised by noting that MA can be factored as

where

and therefore can be written as a two-step scheme. The operation

gives a provisional solution at the vertices. The operation

completes the update by integrating around the vertices. This is in fact the version of Lax-Wendroff known as the Rotated Richtmyer scheme[RIC 62]. It is shown schematically on the left of Figure 3. The original motivations for this scheme were compactness, computational economy and stability. In the nonlinear case, as in all two-step Lax-Wendroff schemes, one avoids any multiplication by the Jacobian matrices. The vorticity-preserving property does not seem to have been previously noticed. It is shown in [MOR 99] that the scheme is stable for the maximum possible CFL range cAt/h < I.

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Figure 3. (left) The Rotated Richtmyer scheme. In the first step, symbolised by white arrows, data from the cells is used to create a half time-step solution at the vertices. In the second step (black arrows) integration round the vertices updates the central cell, (right) Ni's cell-vertex scheme. In the first step (white arrows) we integrate around the cells to obtain a 'cell-residual'. In the second step (black arrows) these are distributed to the vertices. 3.5.

Duality

Since both factors of MA depend only on LA they commute, and so the scheme may also be written as

In this form it is Ni's cell-vertex scheme [NI 82], in which the variables are usually thought of as located at vertices of a grid, defining a bilinear interpolant over a square element. The first step LA is to integrate dxF + dyG over this element and the second step is to distribute this to the nodes of the element. This distribution operation is described by the first factor in (17) 4.

Commentary

By choosing a correct definition of discrete vorticity, the results above can be extended [MOR 99] to the linear wave equation on unstructured two- and three-dimensional meshes, and to problems with non-constant coefficients for which vorticity should be created. In the latter case there is a discrete Kelvin Theorem giving the growth of circulation around a certain class of contour. All of this is of course only a beginning to the design of practical schemes, but an important observation is that the vorticity-preserving property does not depend on choosing any particular expression for the quantity r' in (7). It is enough to compute as a first step any vertex pressure, even a ridiculous one, and then to find the edge presure by averaging along the edges. This gives scope for introducing nonlinear methods (limiter functions) to control the os-

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dilations around Shockwaves without introducing vorticity. It is beyond the scope of this paper to discuss details, but it is possible to design schemes that preserve vorticity in two or three dimensions, and which reduce in one dimension to high-resolution upwind schemes. However, the two-dimensional fluxes are computed from six rather than two neighbouring states, which takes the schemes outside the applicability of the results in [GRE 98]. Therefore schemes that preserve contact discontinuities but avoid carbuncle-type behaviour may still be possible. We are currently investigating such schemes. For the fully nonlinear Euler equations, the evolution of discrete vorticity is quite complicated. However, the evolution of the curl of the momentum is V x (pv) is rather simple, and to avoid unwanted solutions it may be enough to control this quantity. We hope to be able to report shortly on the outcome of numerical experiments. 5.

Bibliography

[MOR 99]

MORTON, K. W.. AND ROE, P., "Vorticity-preserving Schemes of LaxWendroff Type", Submitted to SI AM J. Sci. Comp., 1999.

[ROB 99]

ROBINET, J-CH., GRESSIER, J., CASALIS, G. AND MOSCHETTA, J-M., "Shock Wave Instability and Carbuncle Phenomenon- Same Intrinsic Origin?", preprint, ONERA Toulouse, submitted to J. Fluid Mech., 1999.

[QUI 94]

QUIRK, J. J., "A Contribution to the Great Riemann Solver Debate", Int. J. Num. Meth. in Fluids, 18, pp. 555-574, 1994

[VAN 82]

VAN DYKE, M., An Album of Fluid Motion, Parabolic Press, Stanford, CA., 1982

[PAN 99]

PANDOLFI, M. AND D'AMBROSIO, D., "Numerical Instabilities in Upwind Methods: Analysis and Cures for the "Carbuncle" Phenomenon", preprint, Politechnico di Torino, submitted to J. Comput. Phys., 1999

[GRE 99]

GRESSIER, J. AND MOSCHETTA, J-M., "Robustness versus Accuracy in Shock-Wave Calculations", preprint, ONERA Toulouse, submitted to Int. J. Num. Meth. in Fluids, 1999.

[GRE 98]

GRESSIER, J. AND MOSCHETTA, J-M., "On the Pathological Behaviour of Upwind Schemes", AIAA Paper 98-0110.

[RIC 62]

RICHTMYER, R. D.,"A survey of difference methods for non-steady fluid dynamics", NCAR Tech. Note 63-2, Nat'l. Center for Atmos. Research, Boulder, CO, 1962.

[NI 82]

Nl, R-H.,"A multiple-grid scheme for solving the Euler equations", AIAA Jnl, 20, p 1565, 1982.

On Uniformly Accurate Upwinding for Hyperbolic Systems with Relaxation

Jeffrey Hittinger and Philip Roe Department of Aerospace Engineering The University of Michigan Ann Arbor, Michigan, USA 48109-2140

ABSTRACT The design of uniformly accurate, upwind Godunov schemes for hyperbolic systems with relaxation source terms is discussed. The goal is to develop upwind methods whose sole time step constraints are due to the advection terms yet which obtain accurate solutions even when the relaxation terms are underresolved. An archetypal model system is considered, and analysis of the Riemann initial value problem for this model system is discussed. The ideas learned from this are re-enforced by an asymptotic analysis of general Riemann problems, and these results are described. A strategy for developing a suitable numerical flux function is then outlined. Key Words: hyperbolic systems, relaxation, numerical upwinding, stiff source terms

1.

Introduction

Many flow problems such as those of dilute gases or fluid mixtures have natural formulations as hyperbolic systems with relaxation source terms. These pose interesting challenges for numerical approximation. The source terms cause the system to be dispersive; generally, both the eigenvalues and eigenvectors of the system are modified by the relaxation processes which drive the system towards equilibrium. Another issue arises when the relaxation source terms operate on much smaller time scales than the advection terms; the problem is then said to be stiff. Unfortunately, many problems simultaneously exhibit behavior in both the stiff and non-stiff limits, as well as in between. It is therefore desirable to develop high-resolution numerical algorithms which are uniformly accurate at all scales. For example, if the data are such that

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the advection terms permit a time step large enough to bring the flow into local equilibrium, one would wish to see an accurate solution of the equilibrium problem emerge without resolving the details of the transition. In recent years, there has been much research into such uniformly accurate methods, although a clearly satisfactory procedure has yet to emerge. Pember's early work in the area [PEM 93] identified many properties desirable for Godunov-type schemes for hyperbolic systems with stiff relaxation but did not resolve the relative significance of the frozen and equilibrium wave speeds. Splitting schemes [CAP 97, JIN 96], have been proposed, but general objections to splitting were made in [ARO 96, ROE 93]. These authors proposed characteristic schemes but found no natural way to make them conservative. A conservative method based on an approximate linearization was proposed in [BER 97]. This uses a staggered grid to eliminate the difficulty of resolving the Riemann problem, which no longer has a self-similar solution. However, such schemes may lose definition of linearly degenerate waves, such as contact discontinuities, which are likely to be very important in reactive or relaxing flows. Currently we are attempting to develop an approximate Riemann solver that represents the physics with uniform accuracy and whose computational costs are acceptable. The general behavior of relaxing Riemann problems is now known thanks to the work of Liu and Zeng [ZEN99], although they have concentrated mostly on qualitative behavior of rather general systems. It is complementary to our own work which has involved very detailed analysis of a linear model system which we feel is archetypal. This model has an exact (integral) solution for the Riemann initial value problem, which allows one to see how a simplified solution might be developed.

2.

A Model System

To gain a better understanding of the behavior of hyperbolic systems with relaxation source terms, we have studied the linear1 system

where x € M. and t € R+ are the spatial and temporal independent variables, respectively; e £ K+ is a constant relaxation time; and r e [0,1] corresponds to the equilibrium wave speed. 1 Only the linear problem is considered for the obvious reason that analysis is made tractable. However, this is not the restriction it might seem, as other authors [BER 97] have identified a suitable linearization for nonlinear hyperbolic systems with relaxation. We will demonstrate their requirements in Section 3.2.

The first equation (la) describes the evolution of the conserved quantity u(x,t). The flux of u, that is v(x,t), has its own evolution equation (Ib), and is driven towards equilibrium over a time scale O(e) by the relaxation source term. In equilibrium, this source term vanishes (v = ru), and the system reduces to the single linear advection equation

with the equilibrium wave speed r. Near equilibrium, say v — ru + O(e), it can be shown that

which is an advection-diffusion equation. In similar analysis applied to more complex problems, such as moment approximations of the Boltzmann equations, (2) corresponds to the Euler equations, and (3) to the Navier-Stokes equations. 3. 3.1.

The Riemann Problem The Model Problem

For upwind schemes, the Riemann problem is the heart of the numerical algorithm. We will describe the Riemann problem for the model system and point out behavior that is generic to all relaxation systems. Setting (f)x = —u and (j>t = v allows the system (1) to be reduced to a scalar equation, to which an exact integral solution can be constructed using classical methods. (For details see [ROE 93].) This integral solution is amenable to asymptotic analysis, particularly at small times, and can also be evaluated numerically to provide a complete picture of the evolution of the solution from the small-time to the long-time asymptotics. Consider the system (1) in vector form

with the piecewise constant initial conditions

The discontinuity in value at the origin has a domain of influence bounded by the frozen characteristics |£| = \x/t\ = 1. Outside of the domain of influence, the initial conditions are constant, and, hence, the system of partial differential equations (4) reduces to a system of ordinary differential equations in time. This system is easily integrated to find

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where r — t/e. The conserved variable u remains constant while the relaxation variable v decays exponentially from its initial value to the equilibrium value ru. This rapid adjustment of the relaxation variable forms a temporal boundary layer, as is apparent in the contour plots presented in Figure 1. Within the domain of influence, the solution is

where

with ry* = t — x, (* = t + x, T^o — 7?.(0,0), and the Riemann function

is the product of an exponential and the modified Bessel function of the first kind IQ. Examination of this solution shows that, just behind the frozen waves, the solution decays exponentially to the equilibrium state ahead of each wave. That is, the strengths of the frozen waves are decaying exponentially, as can be seen in Figure 1. The actual decay rates are exp(—(1 ± r)r/2). For small times, r = t/e —> 0 , the solution for u(x,t) is a perturbation about the frozen solution UQ, with the O(n) corrections each a polynomial of degree n in the variable £:

where

with

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Figure 1. Contour plots of the exact solution for the Riemann problem U£ = (1,5)T and UR = (10,3)T for the equilibrium wave speed of r — 0.5 and relaxation time e — 0.01. Left-hand plots show a small-time behavior while the right-hand plots demonstrate a longer-time behavior. The dotted lines \x/t\ — 1, representing the frozen waves, are drawn to emphasize the decay of these waves.

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The piecewise-constant frozen solution is replaced by a smoothly varying solution as the result of decay along and interaction between the characteristics. The diagonalization of (4) explicates this behavior of the characteristics:

Here, v = ~(u + v,u — v)T is the state vector of the frozen characteristic variables. On the right-hand side, the diagonal elements correspond to the previously-noted decay along the characteristics. The off-diagonal elements characterize the interaction between characteristics. For long times, T —>• oo, between the original two waves, the integral factors in TL and TR can be expanded by Laplace's method. The leading-order terms of the solution are

In this limit, the original two waves vanish and are replaced by the equilibrium wave at £ = r. In Figure 1, the rise of the equilibrium wave can easily be seen. As the small-time asymptotic behavior decays, the solution behind the frozen waves tends to the states ahead of the waves except in the narrow region of the equilibrium wave. Here, the solution varies rapidly but smoothly from the left state to the right state. Finally, we consider the flux at £ = 0, which corresponds to a cell interface in an upwind Godunov scheme. For the same Riemann problem as Figure 1, the interface state is plotted in Figure 2 together with the small-time and large-time asymptotic expansions. Note that the interface flux is just (v,u)T'. The flux begins as the frozen flux, then varies smoothly but not necessarily monotonically to the equilibrium flux with this transition in the vicinity of r — 1, where the relaxation and advection are equally important.

3.2.

General Linear Systems

Consider the general linear system in the form (4), where now u G Mm and A,Q € E m x m . We assume that rankQ = (m - n) > 0, so that Q has a null space AA(Q), which is the equilibrium manifold in which the equilibrium solution takes place. Let L0 G M nxm be the row matrix of the left eigenvectors of Q which span AT(Q); similarly, let RQ G R m x n be the column matrix of the right eigenvectors of Q which span Af(Q). It is easy to show that the equilibrium (Euler) equation is

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Figure 2. The. interface state for the Riemann problem UL = (1,5) T and UK — (10,3) T for the equilibrium wave speed of r — 0.5 and relaxation time c = 0.01. The solid line is the numerical calculation of the exact solution, the dotted line is the small-time asymptotic expansion up to terms O(r 4 ), and the dashed line is the leading-order long-time asymptotic solution. and the late-time solution to the Riemann problem comes from the self-similar solution to this problem with Riemann data L 0 U£,,LoUR. In [BER 97], it was proposed to select an average matrix Q(U£,U#) such that Lo(Q)ARo(Q) is the correct Roe matrix for the equilibrium problem. Because they avoided consideration of the early time Riemann problem, they had no need to select any particular linearization of A, but if this is simply a standard Roe linearization, then isolated discontinuities will be correctly captured in both limits. If n — (m — 1), the solution is only stable if the equilibrium wave speeds (the eigenvalues of LoARo) precisely interleave the eigenvalues of A [LIU87]. For this case, an equilibrium wave will grow from the smooth flow between each pair of frozen waves, and simultaneously, this equilibrium wave will be diffused. The frozen waves decay unless they carry a jump lying in A^(Q); such waves will be found in both the early and late time solutions. If n < (m — 1), are no theoretical results, but the general picture of frozen shocks undergoing exponential decay with equilibrium shocks arising from the spaces between them appears to be correct. At early times one can obtain expansions similar to (12). The fc-th frozen wave decays such that

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where \k and r^ are the fc-th left and right eigenvectors of the matrix A, and ct;t(0) is the strength of the initial jump. The solution between the waves is again piece wise polynomial of degree n at order n.

4.

Constructing an Upwind Flux Function

The substitution u = exp{Qr}w transforms (4) into wt + exp{-Qr}Aexp{Qr}wx = 0.

(21)

Integrating this around a cell of a regular, uniform grid with cell-centered data, and translating back to the original variables, one finds

where t1 = t — nAt and f = Au. If the solutions fj±i(t) are known on the interface, this is an explicit formula for the new cell average. The exponential decay in the transient solution is represented explicitly, and within the integral, the exponential factor acts as a filter. To elucidate the weighting in the integrand, let z = Lu, where L is the row matrix of left eigenvectors of Q. This transformation creates a state vector of the n conserved variables (those in JV(Q)) and of the (m — n) relaxation variables which vanish in the equilibrium limit. Multiplying the update formula by L, one obtains

where f = LAL lz and A is the diagonal matrix of the eigenvalues of Q. For conserved variables, the corresponding eigenvalues are zero, so the exponential factors in the update (23) are just unity. For relaxation variables, interpreting At as the time step based solely upon the advection terms, the exponential weighting in the integrand is essentially unity over the entire interval if At/e

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The difficulty is only with those cells for which Ai/e = 0(1). An empirical blending of the two formulae will probably not be very accurate, because of the non-monotone transition seen in Figure 2. However, Figure 2 also shows that the early-time expansion may have a sufficiently large radius of convergence that it can capture the crucial behavior. We are presently investigating such expansions in the general case. The solution for the jumps is given above, and the solution for the linear variation is also straightforward. The quadratic and higher terms appear more complicated, although there is a straightforward means to derive them. But even if the formulae prove expensive they should only be needed at relatively few interfaces. 5.

A Practical Example

Between the continuum (Euler) model of fluid flow and the molecular (Boltzmann) description, there lie many intermediate models. The hierarchy of models devised by Levermore [LEV 96] are of especial interest, because by design, they possess many properties required for trouble-free computation: they are hyperbolic, symmetrizable, entropic, and well-posed. Beyond the Euler equations, the next member of the hierarchy retains ten moments of the Boltzmann equation by assuming that the distribution of the random velocity c is proportional to exp(—^©"^CjCj), where QIJ is a symmetric, non-negative 3x3 matrix closely related to the temperature. Moments of this distribution give the pressure and temperature as tensor quantities. The relaxation process is simply that the temperature tries to relax back toward a scalar,

and the eigenstructure of the frozen problem is straightforward (See [BRO 95]). We feel that although the procedures outlined in this paper may be expensive in general, they will prove relatively simple for well-motivated physical models. 6.

Conclusions and Future Work

The analysis of a simple linear model system has provided clues towards the development of a uniformly accurate upwind method for hyperbolic systems with relaxation. Specifically, the solution of the Riemann problem can be constructed and analyzed for this model system, and this analysis has identified strategies for designing upwind methods for hyperbolic systems with relaxation source terms. Currently, we are implementing and evaluating these ideas.

366

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Acknowledgment

This work is supported in part by a U.S. Department of Energy Computational Science Graduate Fellowship.

8.

Bibliography

[ARO 96]

ARORA, M., Explicit Characteristic-Based High-Resolution Algorithms for Hyperbolic Conservation Laws with Stiff Source Terms, Ph.D. thesis, The University of Michigan, 1996.

[ARO 98]

ARORA, M. AND ROE, P., "Issues and Strategies for Hyperbolic Problems with Stiff Source Terms", Barriers and Challenges in Computational Fluid Dynamics, V. Venkatakrishnan et al., eds., Kluwer Academic Publishers, Norwell, MA, 1998, pp. 139-154.

[BER 97]

BEREUX, F. AND SAINSAULIEU, L., "A Roe-type Riemann Solver for Hyperbolic Systems with Relaxation Based on Time-Dependent Wave Decomposition", Numer. Math., 77, 2, 1997, pp. 143-185.

[BRO 95]

BROWN, S., ROE, P., AND GROTH, C., "Numerical Solution of 10Moment Model for Nonequilibrium Gasdynamics", AIAA Paper 95-1677, June, 1995.

[CAP 97]

CAFLISCH, R., JIN, S., AND Russo, G., "Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation", SIAM J. Numer. Anal., 34, 1, 1997, pp. 246-281.

[JIN 95]

JIN, S., "Runge-Kutta Methods for Hyperbolic Conservation Laws with Stiff Relaxation Terms", J. Comput. Phys., 122, 1, 1995, pp. 51-67.

[JIN 96]

JIN, S. AND LEVERMORE, C., "Numerical Schemes for Hyperbolic Conservation Laws with Stiff Relaxation Terms", J. Comput. Phys., 126, 2, 1996, pp. 449-467.

[LEV 96]

LEVERMORE, C., "Moment Closure Hierarchies for Kinetic Theories", J. Stat. Phys., 83, 5/6, 1996, 1021-1065.

[LIU87]

T.-P. Liu, " Hyperbolic conservation laws with relaxation", Math. Phys, 108, 1, 1987, pp. 153-175.

[PEM 93]

PEMBER, R., "Numerical Methods for Hyperbolic Conservation Laws with Stiff Relaxation: II. Higher-Order Godunov Methods", SIAM J. Sci. Comput, 14, 4, 1993, pp. 824-859.

[ROE 93]

ROE, P. AND ARORA, M., "Characteristic-Based Schemes for Dispersive Waves: I. The Method of Characteristics for Smooth Solutions", Numer. Methods Partial Differential Equations, 9, 5, 1993, pp. 459-505.

[ZEN99]

ZENG, Y., "Thermal Nonequilibrium and General Hyperbolic Systems with Relaxation", preprint, University of Alabama at Birmingham, 1999.

Comm.

Implicit Finite Volume approximation of incompressible multi-phase flows using an original One Cell Local Multigrid method

Stephane VINCENT and Jean-Paul Avenue Pey-Berland BP 108 33402 Talence Cedex France

CALTAGIRONE

ABSTRACT The numerical simulation of multi-phase flows involving stretching and tearing of interfaces requires accurate tools, able to describe near the free surface the different scales of the flow which results from the development of instabilities. On fixed Cartesian mesh, an original local multigrid method, which refines the grid at the cell scale and adapts in time and space, is proposed. An implicit Finite Volume solver, coupled with a TVD- VOF like interface capturing method, is carried out on each grid level. The method is validated and discussed on analytical velocity fields and Rayleigh-Taylor instabilities. Key Words: free surface flows, multigrid method, implicit Finite Volumes

1. Introduction

The numerical simulation of multi-phase flows with strong stresses acting on the interface is classically achieved by the implementation of fixed Cartesian meshes with an interface tracking method (Marker [DAL 67], VOF [YOU 82] or Level Set [SUS 97]). However, due to the memory limit of supercomputers and the computational time, the numerical simulation of non-symmetric threedimensional free surface flows is restricted to problems where the length-scales of the phenomena occurring near the interface are close. To limit the computation node far away from the free surface and concentrate the calculation points on the interface, an original One Cell Local Multigrid method (OCLM) is proposed. Starting on a coarse grid GO which corresponds to the physical domain 17, a refinement criterion Rc is defined to detect the points to be refined on

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each multigrid levels GI, 1 < / < lmax- A hierarchy of embedded subdomains GI,S, 0 < s < smax, is obtained, each fine calculation grid GI>S corresponding to a coarse control volume around a detected point of G/_i cut by 3 in each space direction. An odd cutting ensures a perfect reconnection between the solutions on each GI (Fig. 1). Contrary to the classical multigrid methods such as FIC [ANG 92] or AMR [BER 89], where the fine calculation meshes contains tens to hundreds of cells, all the multigrid calculation domains have the same size (3 x 3) in the OCLM technique, which is an outstanding property of the method. Indeed, on meshes of reduced size, the numerical solvers converges quickly and requires very low memory.

Figure 1. Local mesh refinement technique at the scale of a control volume around a coarse node. After a brief presentation of the motion equations, the numerical solver on a unique grid is explained. The various stages of the OCLM method are next described and finally, the local refinement technique is discussed on classical interfacial problems. 2. 1-fluid model and unique grid solver

The multi-phase flow is modeled by means of the dimensional Navier-Stokes equations for incompressible fluids. In Cartesian coordinates, with a bounded domain fi, we get

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where p is the pressure, u is the velocity field, C is the phase function , g is the gravity, p is the density and p, is the dynamic viscosity. The colour function C repairs the different phases of the flow standing for example C = 0 in one fluid and C — 1 in the other. The interface between these fluids is naturally defined as C = 0.5. The motion equation system (1-3) is a 1-fluid model, in which the discontinuous physical characteristics are estimated according to the arrangement of the fluids as follows

where /?o,pi,/^o and /^i are the physical characteristics of fluids 0 and 1. Finite Volumes on a staggered grid (MAC) are investigated to discretise the Navier-Stokes equation system (1-3). In (1-2), the temporal derivatives are approximated by a Gear scheme of second order, whereas the discretisation of the spatial derivatives is achieved through a Quick scheme for the non-linear terms and a centred scheme for the diffusive one. Moreover, in the presence of discontinuous physical characteristics, a robust augmented Lagrangian numerical solver (Vincent and Caltagirone [VIN 99]) is carried out to calculate the solution of (1-2). The coupling between the pressure p and the velocity u is gone around thanks to a penalization terms added in the momentum conservation equation (Fortin and Glowinski [FOR 82]). Then, the implicit discrete equation system reads

where At is the time scale, r is a numerical parameter controlling incompressibility and n is a normal to F. The exponent n corresponds to time (n At) and subscript 0 refers to the coarse grid GQ. u^ is a reference velocity and B^ is a volume control parameter, which is used to impose a velocity in fi (Angot [ANG 89]) whereas B£ is a Fourier like surface control parameter enforcing boundary conditions on F. An iterative Bi-Conjugate Gradient Stabilised algorithm (BiCgStab, Van Der Vorst [VAN 92]), preconditionned with a Modified Incomplete LU method (MILU) is investigated to solve the linear system gen-

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erated by the discretisation of the motion equations. In the presence of a discontinuous phase function C, the hyperbolic advection equation (3) must be discretised by means of non-oscillating TVD schemes of high order (Vincent and Caltagirone [VIN 99]). In this way, no spurious oscillations appear and a second order accuracy is obtained in each fluid. At the interface, the order of the TVD scheme used is decreased to ensure the monotonicity of the solution. With the implementation of a Lax-Wendroff TVD scheme, accurate solutions were obtained on complex multi-phase flows (see [VIN 99] and [VIN2 99]). 3. One Cell Local Multigrid solver

If it is supposed that the solution (UQ ,PQ , CQ] of (5) on the coarse grid GO at time (n At) is known, then (UQ+I ,p£+l ,CQ+I) can be calculated solving (5) on GO- A refinement criterion Rc is defined to detect the coarse points where a multi-scale solution is necessary. For free surface flows, a local mesh refinement is built on the coarse control volumes cut by the interface. In this way, Rc is expressed as If Rc (M) ^ 0 with M € GI-I, the control volume around M is refined and a fine calculation domain G^s is created. For all I such that I

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Dirichlet boundary condition enforces the numerical solver to converge to a non suitable solution. To relax the stresses at the boundaries of the multigrid calculation grids, composite boundary conditions have been introduced in the Navier-Stokes equation system as follows

A theoretical work was developed to impose the conservation of the shear stress between two multigrid levels (see [VIN3 99]). In this way, orders of magnitude of the composite boundary condition parameters B^ and B^ were estimated, included between 104 and 109. Moreover, the Navier-Stokes solving on the 5 x 5 fine grids is improved thanks to an incomplete ILUD preconditionnning, more powerful than the MILU one.

4. Results

4.1. The vortex test A concentration circle of radius 0.15 meter located at point (0.5, 0.75) in a 1 meter-long square domain is stretched in the symmetrical vortex velocity field of Rider and Kothe [RID 95]. The authors demonstrated that only the Marker method can reproduce the fine scale of the solution. However, this technique is very expensive. All the other methods tested in their paper destroy the fine features of the interface. The Level Set technique and the TVD scheme solution induce strong diffusion and lose precision when the interface is strongly stretched. The PLIC VOF method introduces artificial surface tension in the small scale regions of the interface which are artificially cut. To be efficient, these techniques require very fine calculation grids. Thanks to the OCLM method, coupled with a TVD interface capturing method, the problem is precisely solved with a very low number of calculation points (Fig. 2). Starting with a 70 x 70 coarse grid and applying the OCLM method on two levels, a precision equivalent to 630 x 630 is reached in interfacial regions. Thanks to the local mesh refinement, the memory costs have been divided by 4 with respect to a computation on a full 630 x 630 grid. The local and adaptative character of the OCLM method is perfectly illustrated in this problem.

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Figure 2. Multigrid simulation of the vortex problem. G0 is a 70 x 70 grid. A 3-level solution is presented after large deformations have been induced. The local mesh refinement (only 40% of the total cells are plotted for convenience) and the interface position (C=0.5) on G^ are presented.

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4.2. Rayleigh-Taylor instabilities

Figure 3. Numerical simulation of two-dimensional Rayleigh-Taylor instability by means of the OCLM method. The initial perturbation is 10% of the domain height. The viscosities are the same in the two fluids. The results are presented respectively at time 2.25 s. The distribution of the multigrid cells on the coarse grid and the free surface (C = 0.5) are plotted. The Atwood number A is 0.05 and the Reynolds number Re is 10. The Rayleigh-Taylor instability is a classical and widely studied phenomenon which underlines the competition between the surface tension forces and the viscous stresses. When a heavy fluid lays above a lighter one, any perturbation of the interface between these fluids is amplified under the action of gravity,

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whereas the surface tension tends to minimise the deformation of the free surface. The Rayleigh-Taylor instability problem is simulated to highlight the abilities of the OCLM method to solved free surface flows. The Navier-Stokes equations and the interface capturing procedure are computed on all the grid levels. For example, the local mesh refinement is carried out on 2 grid levels (Fig. 3). GO is a 40 x 80 grid. The free surface evolutions are perfectly captured by the OCLM method. The local character (4 or 5 cells surrounding the interface) and the adaptative property of the method in time and space are verified. The composite boundary conditions have allowed to solved precisely the motion equations on the fine calculation domains. Moreover, the reconnection between the hundreds of fine grids is perfectly achieved thanks to the odd cutting. In the present problem, the ratio between the number of calculation point in the OCLM solving and the number of node on an equivalent fine 120 x 240 grid is always less than 0.09. The gain in memory is so very important and the calculation time is decreased about 10 %. The difference between the memory and calculation time improvements lays in the additional operations needed to solved on the multigrid levels and in the detection procedure.

5. Conclusion

We have presented a new local and adaptative multigrid refinement method (OCLM) for the numerical simulation of free surface flows. The motion equations and the interface capturing are solved at the cell scale thanks to the definition of new composite boundary conditions. The method brings important gains in memory and calculation time. The extension of the method to three dimensions is immediate. Moreover, the OCLM method is a general method independent on the equations solved as well as on the numerical solver. The method was successfully applied to the numerical simulation of natural convection in a square cavity. [DAL 67]

DALY B. J., Numerical study of two-fluid Rayleigh-Taylor instability , Phys. Fluids, 10, 1967, p. 297-307.

[YOU 82]

YOUNGS D. L., Time-dependent multi-material flow with large fluid distortion, K. M. Morton and M. J. Baines, 1982, p. 27-39.

[SUS 97]

SUSSMAN M. AND SMEREKA P., Axisymmetric free boundary problems , J. Fluid Mech., 341, 1997, p. 269-294.

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[ANG 92]

ANGOT P., CALTAGIRONE J.-P. AND KHADRA K., Une methods adaptative de raffinement local: la correction du flux a 1'interface , C. R. Acad. Sci. Serie II b, 315, 1992, p. 739-745.

[BER 89]

BERGER M. J. AND COLLELA P., Local adaptative mesh refinement for hyperbolic partial differential equations , J. Comput. Phys., 82, 1989, p. 64-84.

[VIN 99]

VINCENT S. AND CALTAGIRONE J.-P., Efficient solving method for unsteady incompressible interfacial flow problems , Int. J. Numer. Methods Fluids, 1999, to be published.

[FOR 82]

FORTIN M. AND GLOWINSKY R., Methodes de Lagrangien augmente. Application a la resolution numerique de problemes aux limites, Dunod, 1982.

[ANG 89]

ANGOT P., Contribution a 1'etude des transferts thermiques dans des systemes complexes: application aux composants electroniques, PhD thesis, University Bordeaux I, 1989.

[VAN 92]

VAN DER VORST H. A., Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems , SIAM J. Sci. Stat. Comput., 13, 1992, p. 631644.

[VIN2 99]

VINCENT S., CALTAGIRONE J.-P. AND ARQUIS E., Numerical simulation of liquid metal particules impacting onto solid substrate: description of hydrodynamic processes and heat transfers , J. High Temperature Material Processes, 1999, to be published.

[CAL 95]

CALTAGIRONE J.-P., KHADRA K. AND ANGOT P., Sur une methode de raffinement local multigrille pour la resolution des equations de Navier-Stokes , C. R. Acad. Sci. Serie II b, 320, 1995, p. 295-302.

[VIN3 99]

VINCENT S. AND CALTAGIRONE J.-P., One Cell Local Multigrid method for solving multi-phase flows , J. Comput. Phys., 1999, in corrections.

[RID 95]

RIDER W. J. AND KOTHE D. B., Stretching and tearing interface tracking problems , AIAA paper, 95, 1995, p. 1717.

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New Classes of Integration Formulas for CVFEM Discretization of Convection-Diffusion Problems E.P. Shurina and T.V. Voitovich Novosibirsk State Technical University, Novosibirsk, Russia ABSTRACT This paper is concerned with the development of FEM-like finite volume technique for discretization of convection-diffusionn problems over unstructured simplicial grids. Proposed finite volume technique is original in it's essence - technology of approximation of convection-diffusion fluxes and sources. Polynomial interpolation is used to represent unknown variables, ttransfer coefficients, thermophysical characteristics and source terms as linear combinations of local form functions at each element. Local form functions are expressed in terms of simplex barycentric coordinates. Three special classes of non-symmetric formulas for exact integration of monomials of barycentric coordinates over dual mesh lines and corresponding subregions are introduced. The first class contains formulas for exact integration of barycentric coordinate monomials along dual mesh lines lying on a simplex, and is used to express convection and diffusion fluxes in terms of discrete unknowns. The second class consists of formulas for exact integration of monomials of barycentric coordinates over simlplex subregions, corresponding to different finite volumes sharing the same element, and is used for discretization of source terms. The last class prescribes exact integration of the monomials along segments of the edges, approximating boundary curves, and is intended for realization of boundary conditions. Geometric interpretation of proposed integration formulas of the lowest order is presented. Described technology can be applied to a wide variety of PDE models, and with the use of volume barycentric coordinates allows uniform extension to three-dimensional case. Key Words: control volume finite element methods; unstructured grids; dual mesh; barycentric coordinates; upwind schemes.

1.

Introduction

Adequate numerical simulation of fluid and gas dynamics processes imposes strict requirements on the modern methods, namely: exact representation of domains with complex geometry and high-order accuracy of spatial and temporal discretization schemes. Two types of grids, non-orthogonal and unstructured allow

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exact treatment of complex domains. We use the latter to account for additional requirements, such as adaptivity and local grid refinement realization. As for the spatial discretization, the finite volume method has some advantages, in particular, conservative properties of corresponding schemes, simplicity and ability of more natural use of the upwind schemes. Formulation of a finite volume element scheme typically involves the following basic steps 1. A set of admissible discretization templates is chosen. 2. An arrangement of all unknowns of source nonlinear coupled system (cellvertex, cell-centered, side-centered arrangement) is specified. 3. Element based interpolation polynomials are specified for transfer coefficients, conservative variables, thermophysical variables and source terms. 4. Weighted residuals principle, with weighting functions being equal to one over each finite volume is applied. 5. For each element convection-diffusion fluxes through the dual mesh lines are expressed as linear combinations of local discrete unknowns, with coefficients of the combination being components of the local diffusion and convection matrixes. Similarly, integration of the source term interpolation polynomial over subregions of a finite element gives the local source vector. 6. Global matrixes and vectors are obtained via the element-by-element assembly. 7. Resulting sparse linear system is solved by a specified method. We are interested in the development of a finite-element like procedure for generation of discrete analogue coefficients, thus we will take a closer look at the fifth step. The main contribution of this paper lies in the development of a new uniform technique of convection-diffusion fluxes and source terms approximation, with exact integration of interpolation polynomials, which are represented with simplex barycentric coordinates. The technique is based on the three classes of special formulas introduced by the authors. These classes give exact values for integrals of barycentric coordinates monomials along dual mesh lines, simplex subregions and triangle edges segments. Alternative approach with exact integration of polynomials in finite volume discretization, in which all polynomials are expressed as a generalized Taylor series, was proposed by Yen Liu and Marcel Vinokur [LIU 97]. In FE discretization of Navier-Stokes equations only a few convergent pairs for pressure-velocity interpolation functions are known. In the case of triangular finite elements linear pressure - quadratic velocity is a usual choice. It has been shown that the Ladyzhenskaya-Babuska-Brezzi (LBB) restrictions still hold for hybrid FEM/FVM techniques [SHU 97]. To our best knowledge, appearance of LBB restrictions strongly depends on the choice of a pressure-velocity coupling technique. For example, SIMPLE-like algorithms [PRA 85] for linear interpolation functions of equal order do not lead to ill conditioned matrices for the pressure correction. Finite element community also reports several methods of circumventing the LBB restrictions [ZEN 95]. Thus, the triangular based linear form functions can be used to demonstrate the main features of the new approximation approach. However, the form functions of higher degrees are allowed, and this extension will be discussed later.

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2. Proposed finite volume technique Consider the conservation equation for a scalar quantity 0

where p is the fluid density, u is the velocity vector, F^ is the diffusion coefficient and SQ is the volumetric source of . Domain discretization. The computational domain is divided into triangular elements. Assume that computational points and finite element vertexes coincide ("the cell-vertex" arrangement), and both velocity components and scalar variables have the same finite volume ("the collocated" arrangement). Dual mesh is constructed on centroids of triangles and mid-points of the edges, such that each node / has the corresponding complete, or "incomplete" finite volume Q, , bounded by median segments (see Figure 1). Let Sf denote the boundary of £2,. Let us assume fixed global and local numeration on the set of triangulation nodes and on each finite element. The integral conservation form of equation (1) is

where J = Jc + Jd is the combined convection-diffusion flux of <j). The proposed CVFEM technology, similarly to the known ones, uses the following principles of finite element discretization: (i) specification of elementbased form functions for conserved quantities and sources (it should be noted, that the usage of interpolation functions by FV is more flexible, because different classes of profiles can be applied to different terms of equation (1)); (ii) element-by-element assembly of discrete analogue matrices and vectors. The essence of our contribution is the use of barycentric coordinates in local representations of scalar variables,

Figure 1. Fragment of primary (FE) and secondary (FV) grids corresponding to linear interpolation and notation used for a triangle en .

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transfer coefficients and sources and introduction of special integral formulas for barycentric coordinates monomials, much as in FEM. Consider a finite element en. Let 5"^ denote a segment of the dual mesh, lying on a median, which comes out of the triangle node nb (see Figure 1), and Jv nfr is a value of the combined flux through the median segment S^b in the direction of the outward normal to the boundary of the finite volume £lv , (see Figure 2). It will be sufficient to approximate three of the six introduced fluxes, because Diffusion

fluxes approximation.

Using Green's formula, the diffusion flux

through a median segment S^ can be represented as follows

It is well known that FEM naturally allows representation of significantly varying transfer coefficients by an interpolation function on an element, unlike most of the FVMs. Let us demonstrate proposed technology on a problem of approximation of the diffusion flux, using linear interpolation for F^ and 0 [3]:

Figure 2. Outward unit normals and corresponding notation for fluxes.

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where \j/m - iff m (x) are local linear basis functions (see Figure 3), Tm are the local values of F^ (m=l,2,3), and coefficients a^,b^,c^ can be expressed in terms of the local discrete unknowns 0i,02>03 coordinates of the triangular nodes:

va

'

an

mver

se matrix of the matrix of

[6]

Thus, approximation of the diffusion flux [3] becomes

It is clear from [7] that it is necessary to introduce special formulas for integration of the local basis functions along the median segments to complete discretization of the diffusion flux. Note, that in the case of triangular based linear interpolation, local basis functions coincide with simplex barycentric coordinates L | .,^ | .(x)=L | ., i = l,2,3. For every local basis function consider a pair of trapeziums and a triangle, which are constructed using finite element median segments and their images on the surface of the basis function (see Figure 3). Projections of the trapeziums and the triangle to coordinate planes gives

Figure 3. Some notation for the median projections and one of the local linear functions on the triangle en.

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Finite volumes for complex applications

Here lvx , /^ are the lengths of the projections of the median segment 5" onto the x and y axis, respectively, with the signs corresponding to the anticlockwise integration; v,nb& {1,2,3}. Taking into consideration formulas [8],[9] , we have for three determining diffusion fluxes of the element:

Finally, to determine coefficients in the first, second and third row of the local diffusion matrix, combinations 7^ - ^23' ^ 2 3 ~ ^ 3 l > ^31~^12 are considered in terms of conservation unknowns. Coefficients at 01? 02» 03 *n tne combinations give three elements in the current row of the local diffusion matrix. Approximation of convection fluxes is performed as

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383

Special Prakash-Patankar interpolation of u is used to avoid spurious pressure oscillations in the framework of the collocated approach. An upwind scheme is used to express (f)nb in terms of the local unknowns. The flux f „ pu-nds is \j> approximated using formulas [8,9]. Source terms approximation. Let Q.n, denote an intersection of the finite element en of interest and the finite volume of the node with local number nb

Figure 4. Notation for the finite element subregions Formulas for exact integration of barycentric coordinates over the subregions are introduced as follows

(One can easily obtain [14] representing each of the calculated volumes as a set of truncated right prisms). Formulas [14] are used to compute local RHS vector of en . Boundary conditions. For realization of boundary conditions (f.e. approximation of convection fluxes through outlet boundaries), one-dimensional analogues of [8,9] for integration of the local basis functions along the element edges segments are introduced. Future investigations. It is well known in the finite element community, that an advantage of the use of barycentric coordinates, is in the existence of integration formulas, simplifying computations of integrals along the edges of a triangular element and over an element itself, namely the following formulas by Eisenberg and Malvern[EIS73]:

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It is obvious, that in the proposed FV technology formulas [8,9,14] play the same role as [15,16] in FEM, and have the lowest order. Following classes of formulas for exact integration of barycentric coordinates monomials (integration over dual mesh lines, finite element subregions and boundary edges segments, respectively) are the subjects of our investigation:

Classes of the formulas [17] are the basis of the uniform discretization technique, which allows to form functions of higher order, since these functions can be represented through barycentric coordinates. Although the capabilities of [8,9,14], are such that these formulas of the first order can be used as a basis for CVFEM discretization, we consider the technique incomplete until the exact values for [17] are be found.

3.

Concluding remarks

A new approach for finite volume discretization is developed in analogy with the FE approach, namely we use barycentric coordinates in local representation of scalar variables, transfer coefficients and sources. Final discrete analogues of diffusion-convection fluxes and sources are obtained via special classes of integration formulas introduced by the authors.

4.

Bibliography

[LIU 97]

Liu Y., Vinokur M. Exact integrations of polynomials and symmetric Quadrature Formulas over Arbitrary Polyhedral grids, J.Comput.Phys., 140, 122, 1997.

[SHU 97] Shurina E.P., Voitovich T.V., Analysis of the finite element and finite volume methods based upon unstructured grids for solution of the Navier-Stokes equations, Computational Techlologies, 2,4, 1997 (in Russian). [PRA 94] Prakash, C, Patankar, S.V., A control volume-based finite-element method for solving the Navier-Stokes equations using equal-order velocity-pressure interpolation, Numer.Heat Transfer, 8, 259, 1985. [ZEN 95] Zienkievicz, O.C., Codina, R., A general algorithm for compressible and incompressible flow. Part I: The split, characteristic based scheme', Int.j.numer.meth. fluids, 20, 869, 1995. [EIS 73] Eisenberg M.A, Malvern L.E., On finite element integration in natural coordinates, Int.j.numer.methods eng., 1, 574, 1973.

Fields of application

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Analysis of Finite Volume Schemes for Two-Phase Flow in Porous Media on Unstructured Grids

M. Afif (1) and B. Amaziane(2) (1) Universite Cadi Ayyad Faculte des Sciences Semlalia B.P. S.15, 40000 Marrakech Maroc. E-Mail: [email protected] (2) Universite de Pau et des Pays de I'Adour Departement de Mathematiques Recherche Av. de I'Universite, 64000 Pau France. E-Mail: [email protected]

ABSTRACT This paper is devoted to the analysis of finite volume schemes on unstructured grids for a nonlinear, degenerate, convection-diffusion equation arising in flow in porous media. A semi-implicit scheme is considered. We prove that this scheme is L°°, weak BV stable under a CFL condition and satisfy a discrete maximum principle. We then derive a convergence result. Results of numerical experiments using the present approach in the 2-D case are reported. Key Words: finite volume method, degenerate parabolic equation, nonlinear diffusion-convection, porous media, unstructured grids.

1. Introduction Flow simulation in petroleum and groundwater reservoirs has been extensively studied using finite element methods in past years (see, e.g., [AWZ 96, CHJ 86, CHE 97] and the bibliographies therein). Also, a discretization using both finite element and finite volume methods for two-phase flow in porous media is presented in [CJR 95]. However, it appears that there are few results on convergence theory for the degenerate problem [AWZ 96, CHE 97]. More

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recently, finite volume methods were developed and analyzed for immiscible two-phase flow in porous media in the case where the diffusion term is neglected, (see [EGH 98, VEV 98] and the references therein). This approach leads to robust schemes applicable for unstructured grids and the approximate solution has various interesting properties which correspond to the properties of the physical solution . These methods have been useful for advective flow problems because they combine element by element conservation of mass with numerical stability and minimal numerical diffusion. In this paper we will study a numerical approximation of two-phase flow in porous media. We focus on two-phase immiscible flow, which corresponds physically to the water flooding of a petroleum reservoir. We consider twophase water and oil flow in a porous media, using the global pressure, the total velocity and the water saturation as the primary variables. This formulation leads to a coupled system of partial differential equations which includes a nonlinear degenerate parabolic saturation equation and an elliptic pressurevelocity equation [CHJ 86]. The saturation equation is convection dominated and thus special care should be taken in discretization. The diffusion term there is small but important and can not be neglected. Also irregular geological features, which suggest the use of irregular grids. We will consider a system of equations describing the flow of two immiscible incompressible fluid phases through a porous medium in which for sake of simplicity, we will neglect the effect of gravity. This system is then the following: Saturation Equation:

Pressure Equation:

where u(x,t) is the water saturation, P is the global pressure, K(x) is the absolute permeability tensor of the reservoir Q C R2 whose boundary F splits up into three parts such that F = FI U ^ U Fa and F,- D Fj = 0 for i ^ j; FI is the part of the boundary where the water is injected, F2 is the impervious part of the boundary and T3 is the producing part of the boundary. Also Qr denotes £l x ]0, r[.

Fields of application

389

the saturation equation that satisfy the discrete maximum principle and derive convergence results. More precisely, we will consider a numerical approximation of this system where the saturation and pressure equation are decoupled. We will concentrate on the study of the convergence of the FVS for the saturation equation taking into account the diffusion term and the anisotropic heterogeneity of the reservoir. A mixed-hybrid finite elements method is used to obtain an accurate approximation of the velocity [BRF 91]. In [AFA 97], the one-dimensional problem was analyzed. In this paper, we extend these ideas to multiple dimensions. We will restrict attention to R2, however, the methodology and the analysis can be extended to problems in R3. This paper is organized as follows. In the next section, we present the finite volume discretization of the problem (P1). An explicit approximation of the convection term combined with an implicit approximation of the diffusion term is considered. The solution is approximated by a piecewise constant. In section 3, we present the L°°, weak BV stability under a CFL condition and a convergence result. Numerical simulations for a 2-D example are presented in section 4. 2. Finite volume discretization

Before describing the finite volume discretization of the model problem (P1) we give some notations. As usual, let ( t n ) n = Q Nr be a partition of J with a time step At = i n + i — tn and let A/j = (Ti)i=o Ne be an admissible triangulation of fi, S/j = (Mi)i=00NS the Donald dual mesh, whereere 2 h := min|M;| verify sup|M,-| < S.h2, and XT the barycenter of T E Ah, is i

i

such that XT -

f| MnT^0

dM £ T. We denote by XM :=

U

dT £ M.

TnM^0

l := dMi U dMj Pi T the line segment between the points XT and Xij the midpoint of (XM, ,XMj) and let £h — {I £ dM\F , forM £ Eh}- For an initial condition u° 6 L°° (Q) n BV (ft), we set u°M = rj^ fM u° (x) dx. For simplicity we assume that the functions 3> and K are piecewise constants, we set $M — *^|M and KT = K\T- Let u^ be an approximation of u ( x M , t n ) which will be defined precisely in the sequel Integrating (1) over the set M x [t n ,tn+1] where M £ Eh,, we obtain the following scheme

where nM,I is the outward normal to / £ dM. Using an explicit approximation of the convection term and an implicit approximation of the diffusion term, we

390

Finite volumes for complex applications

get

which could be written in the following form

where

with MI E Eh is such that / € MI D M. The advection term is approximated by an upwind Godunov scheme and the diffusion term is approximated in the following way: for T PI M = 000

where L £ dT such that LDM = 0, and using a piecewise linear approximation for Va, we get

where XM 6 P1 such that XM,( x Mj) = &ijFinally, since divq = 0 we have the following semi-implicit scheme

where DM,i = - \T| VXM,,T • AWxM.T for all / G dM\T.

Fields of application

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Remark 1 We can use an explicit fresp. implicit] approximation for both the diffusion and the convection terms to obtain an explicit [resp. implicit] finite volume scheme which lead generally to large CPU time computing. For more details on the comparaison of these schemes see [AFA 97].

3. L°° stability weak BV estimates and convergence results In this section we shall analyze the F.V. scheme obtained in section 2. Let us state the following assumptions: is a bounded open polygonal subset of R2. such that - (A3) K is a bounded, uniformly positive definite symmetric tensor on Q

- (A4)

such that

- (A5)

such that

- (A6) b,d

and

such that 6 is a monotone increasing function, and

It should be noted that a full tensor for the absolute permeability K is considered and the approximation obtained in (4) is such that 0 < D_ < DM,i < D+ < oo which is an important property for the analysis of this FVS. In fact, we can write the coefficient DM,I in the following forme:

where L and L' G &T such that L D M = 0 and L' U MI - 0, hence

Furthermore, we have DM,I = -^^X'XM • adj [KT]X'^MI where x' — L fl L' and adj is the adjugate matrix. Now adj [/\T] is also a symmetric definite positive matrix, then it has a unique factorization, adj [KT] — \Ur] [UT], in which UT is upper triangular with a positive diagonal. We have

then for K(x) = k(x)Ko in £7, where k(x] > 0 and Ko is a constant symmetric rrt positive definite matrix such that adj [Ko] = [Uo] [Uo], we can triangulate

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Finite volumes for complex applications

QKo := [U0] (£2) a transformation of the domain fi by the matrix Uo, with an admissible triangulation such that

where TJ is a small parameter independent of h, then we get also 0 < D_ < DM,l • The extension of the present technique to the case where

such that

K(x)|n, p = k p ( x ) K p , with kp (x) > 0 and Kp is a constant symmetric definite positive matrix, for all p = 1,.., Np, is straightforward. Let Cq defined by

where NM0 = card {l E 9M}, such that sup N0M is assumed to be finite, we M introduce the CFL condition defined by:

The following results hold: Proposition 1 Under the assumptions (A1)-(A7) and the CFL condition (5) the scheme (4) is L°° stable. Moreover, the approximate solution (unM} satisfies

Proposition 2 Under the assumptions (A1)-(A7) and the CFL condition (5) with [CFL < I — e], we have the following estimates

where s is a small parameter and C is a constant independents of h and At. Let us introduce a weak solution u of the problem (P1):

The data are assumed to be smooth enough to guarantee the existence and uniqueness of the weak solution. We have the following result: Theorem 1 Under the assumptions (A1)-(A7) the approximate solution Uh, given by the scheme (4) and the CFL condition (5) with [CFL < 1 — e\, converges to u in L2(QT) as h and At goes to zero.

Fields of application

Figure 1. Reservoir &

Figure 2.

393

Triangulation of QQ = UQ (&}

5. Numerical simulations

In this section, we present some numerical results in 2-D. An IMPES simulator which applies a mixed-hybrid finite element method for computing pressure and fluid velocity approximations and the finite volume scheme described here for the saturation was developed. Numerical results prove the effectiveness and the robustness of this methodology. We have run various simulations, herein we are presenting only one of such a simulation to illustrate our results. A simulation was performed on a heterogeneous reservoir shown in Figure 1 for a two phase flow problem. An injection well is placed at the lower left hand corner of the reservoir and a production well is placed at the top right hand corner of the reservoir. The highest value of permeability (K = 10) is in the darkest shaded blocks. The lowest permeability (K = 0.1) is in the lightest shaded blocks. The intermediate value of permeability (K — 1) is the intermediate shade in the picture.

Figure 3. Saturation contours

Figure 4. Total velocity q

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Finite volumes for complex applications

6. Bibliography [AFA 97]

AFIF M. AND AMAZIANE B., On convergence of finite volume schemes for one-dimensional two-phase flow in porous media, Preprint, 1997, to appear.

[AWZ 96]

ARBOGAST T., WHEELER M.F. AND ZHANG N.Y., A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Num. Anal., 33, N° 4, 1996, p. 1669-1687.

[BRF 91]

BREZZI F. AND FORTIN M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.

[CHJ 86]

CHAVENT G. AND JAFFRE J., Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986.

[CJR 95]

CHAVENT G., JAFFRE J. AND ROBERTS J.E. , Mixed-hybrid finite elements and cell-centred finite volumes for two-phase flow in porous media, in: A. Bourgeat, et al., eds., Proceedings Mathematical Modeling of Flow Through Porous Media, World Scientific, London, (1995), p. 100-114.

[CHE 97]

CHEN Z. AND EWING R.E., Fully-discrete finite element analysis of multiphase flow in ground water hydrology, SIAM J. Num. Anal, 34, N° 6, 1997, p. 2228-2253.

[EGH 98]

EYMARD R., GALLOUET T., GHILANI M. AND HEREIN R., Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes, IMA J. Num. Anal., 18, 1998, p. 563-594.

[VEV 98]

VERDIERE S. AND VIGNAL M.H., Numerical and theoretical study of a dual mesh method using finite volume schemes for two phase flow problems in porous media, Numer. Math., 80, N° 4, 1998, p. 601-639.

A preconditioned finite volume scheme for the simulation of equilibrium two-phase flows

Sebastien Clerc Commissariat a I'Energie Atomique, CEA-Saclay, 91191 Gif-sur-Yvette FRANCE

ABSTRACT In this paper, we report on a preconditioned finite volume technique to simulate two-phase flows in complex geometries. Key Words: two-phase flows, low Mach number flows, preconditioning.

1. Introduction

The numerical simulation of two-phase flows is a crucial problem for the design and safety analysis of heat exchangers, especially in the nuclear industry. Generally, the fluid is in the liquid phase at the inlet of the exchanger. Because of the heat release or a loss of pressure, vapor is created. The description of the interface between the phases would require a very fine space scale with respect to the characteristic length of the geometry. For this reason, one must consider a macroscopic model. This model should reproduce some of the characteristic features of two-phase flows. The most basic of these features is probably the compressibility effect introduced by vapor bubbles. Indeed, although the density of liquid water can usually be considered as a constant, the apparition of bubbles creates large variations of the mean density of the flow. Moreover, the dependence of this mean density with respect to the pressure can not be neglected. In other words, if we consider the bubbly flow as a homogenized mixture, the resulting averaged speed of sound is quite low in usual conditions. In order to study this feature, we consider the very simple homogenizedequilibrium two-phase flow model. Indeed, using more sophisticated models would introduce other difficulties which are not in the scope of this study.

396

Finite volumes for complex applications

2. The equilibrium two-phase flow model

The homogeneous equilibrium model of liquid/vapor flows is characterized by a kinematic and thermodynamic equilibrium assumption. The evolution of the mixture can be described by the Euler equations for a single fluid:

Here E = e + |u| 2 /2 denotes the total energy, and H = h + |u| 2 /2 the total enthalpy of the mixture. To close the system, an equation of state (EOS) links the pressure p to the conservative thermodynamic variables p and pe. The pressure law must be such that the partial derivatives x and K with respect to p and pe satisfy The sound speed c of the fluid is the square root of this quantity. Generally, a drift flux is added to the set of equations (l)-(3) to incorporate the effects of the slip velocity between the phases. This will however not be the case in the present study. 2.1 Equation of state The equation of state consists of three zones: a liquid, a two-phase mixture, and a vapor zone. The liquid and the vapor zones are described by usual single fluid equations of state. In the two-phase zone, each fluid is supposed to be at saturation, so that the density pt,pv and the enthalpy hi,hv of each phase are functions of the pressure only. The density p and the enthalpy h of the mixture satisfy :

The transitions between the mixture zone and the two others occur along the saturation curves. The equation of state is continuous across this curve but not continuously differentiable. This fact is clearly visible in Fig. 1.

Fields of application

397

Figure 1: Specific volume l/p versus enthalpy h for the water equilibrium EOS, at pressure p = IMPa. The liquid zone corresponds to lower values of h, and the vapor zone to the higher values of h. In the two-phase zone in-between, 1/p varies linearly with h. Note that the derivatives of the equation of state are discontinuous across the transitions. 3. The preconditioned finite volume scheme 3.1. Some notations

We rewrite the Euler equations (l)-(3) with the compact form:

We consider a triangulation of the computational domain by polygonal cells. For a cell A', we denote by |K | its volume and N ( K ) the set of neighboring cells. If J E N ( K ) , the surface of the common interface is (TKJ and *IKJ denotes the unit normal, oriented from K to J. Finally, we denote by 6t the step of the time discretization. The finite volume method for the solution of the system of conservation laws (4) takes the form:

Here, U% denotes the average value of the solution in the cell K, at time nSt. To completely determine the numerical method, we have to specify the expression of the numerical flux <&K J in terms of the cell average values. We will consider

398

Finite volumes for complex applications

numerical flux functions of the form:

The viscosity matrix Q depends on the left and right states UL and UR. The numerical flux we have used is a preconditioned version of Roe's numerical flux. We will first begin by describing the extension of Roe's flux to our system, and then introduce the preconditioning. 3.2. Roe's flux Roe's scheme is obtained by chosing for Q = |A|, where A is the Roe matrix, and |A| = ^ |A,-|r,-(g)l,-, if A = ^ A,Tj<S>l,- is the eigen-decomposition of A. We will denote by ([•] the difference between the right and left values of any quantity, i.e. (•)# — (-}L- The Roe matrix has to satisfy the following properties: • An has a complete set of real eigenvectors • A n (t/L, UR) -)• DF(U) • n when UL and UR tend to U. • An(ULtUR)lU] = [ V ( U ) - u ] . As is well known (see e.g. [ViMo, To 91]), a Roe matrix for the Euler equation with an arbitray EOS can be found if we can provide a linearization of the pressure jump such that

Geometrically, the solutions to this problem are located on the intersection of a line and a half plane in the (x, K) plane. We will denote by [x] the vector ([/?], fl/>e]) and by Dp = (x, K), such that (7) becomes [p] = [x] • Dp. Following the lines of [ViMo], we start with a point Dp which is some average value of DPL and D p R , but does not satisfy (7). We project it on the line, with respect to a scalar product defined by a given symmetric matrix M:

We can then check a posteriori if condition (8) is satisfied. If not, we must consider another initial guess Dp. It is clear that if p is continuously differentiate, £ stays bounded when [x] tends to zero. Moreover, if we choose for M_an approximate value of the Hessian of p, then £ will be close to one. Finally, Dp will tend to Dp as expected. For

Fields of application

399

our simulations at p = 5 MPa, we found that a diagonal matrix diag(l, 10~ 12 ) gave satisfactory results. However, if we are close to a point where p is not continuously differentiable, then £ will certainly blow up when |x| goes to zero. As a result, the value of Dp will not tend continuously to Dp. This fact lies at the root of some numerical difficulties in the computation of phase transition inside the flow. It becomes particularly important when the discrepancy between the derivatives is large, i.e. for lower values of the pressure. Obviously, more research seems in order on this point.

3.3. The preconditioned flux For low Mach number applications, it is preferable to use a preconditioned version of Roe's scheme. The viscosity matrix (see eq. (6)) is now Q = P-1 |PA|, where P is a properly chosen non-singular matrix. Note that since we have only modified the viscosity matrix, the consistency of the numerical scheme is not affected. Therefore, the scheme is time-accurate and suitable for transient problems. This formulation was first introduced by [GuiVio] and further studied by [Cl 98], see also [PCV+]. In this work, we have used Turkel's diagonal preconditioner [Tu 87]. This matrix depends on a single parameter B which was set to the local Mach number.

4. Implicit time stepping

4.1. Choice of the variables for the linearization In low Mach number applications, an implicit time stepping cannot be avoided. First, the CFL constraint on the time step for explicit schemes becomes too restrictive when the sound velocity gets large. Moreover, for the time-accurate preconditioned scheme, the stability condition is even more restrictive than the CFL condition. One can consider the usual linearized backward Euler scheme:

with It is also possible to linearize the problem with respect to an other set of variables: setting U — U(V), one can consider

400

Finite volumes for complex applications

with Vn+1 = Vn + 6V. At steady state (6V = 0), this formulation is still conservative. The conservation is not ensured during the transient but can be restored if necessary by performing a few Newton steps within a time-step. Note that this is also true of the conservative variables formulation (9). More specifically, we advocate the use of the following set of variables: ( p , p u , p ) . Including the pressure in the set of variables has two advantages in our case. First, it makes the computation of the thermodynamic variables simpler and faster. If the density and the pressure are known, we can readily know in which zone of the EOS the state is located. On the contrary, if the energy rather than the pressure is known, we must first guess in which zone it is, and modify this guess if necessary. More importantly, it can dramatically improve the convergence of steady state computations when contact discontinuities are present (see [Cl 99]). Indeed, the pressure time-increment is usually very small in low Mach number flows, although the density and the energy time-increments can both be important. With a non-linear EOS such as the equilibrium EOS, the conservative variables formulation (9), the density and energy increments will automatically induce spurious large pressure increments. This phenomenon is responsible for the apparition of spurious acoustic waves of large amplitude in the solution, which prevents the use of large time steps in the implicit scheme. With formulation (10), larger time steps can be used. Although in some cases the time step can be set close to infinity, it is generally advisable to stay close to 6x/u, where 6x is the typical mesh size and u is the reference material velocity. In low Mach number flows, this time step is of course much larger than the maximum explicit time step. 4.2. Approximate Jacobian for the preconditioned flux Equation (9) (and (10)) require the computation of the partial derivatives dO/dUL and dQ/dUR of the numerical flux. Due to the algebraic complexity of the numerical flux, an exact computation of these matrices is not easy, especially for a general EOS. We have thus employed the usual approximation for Roe's flux:

To obtain the derivatives with respect to V, we left-multiply these matrices by the Jacobian of the change of variables dUfdV. Practically, it amounts to expressing the left eigenvectors of A (for Roe's scheme) or PA (for the preconditioned scheme) in terms of the pressure rather than the energy.

Fields of application

401

5. A two-phase bump channel flow.

We briefly present a two-dimensional test-case for the equilibrium two-phase flow model. We consider a channel with a 20% sinusoidal bump (Fig. 2). A

Figure 2: Channel with bump: 20 x 80 structured mesh. pressure of 5 Mpa is imposed at outflow. A constant velocity of 10 m • s-1 is imposed at the inlet, and the enthalpy is set to 1154.2 KJ. This corresponds to a liquid state close to the saturation. As the pressure drops with the restriction of the section, a small concentration of vapor appears. The Mach number jumps from 10- 2 to 0.4, so that the compressibility becomes important. We initialize the computation with a purely liquid steady-state computed with a higher value for the pressure. The pressure profile is displayed on Fig. 3. As expected, no discontinuity can be detected across the phase transition lines. On the contrary, the influence of the vapor is clearly visible on the Mach number profile (Fig. 3, right), which shows sharp discontinuities. For both profiles, the symmetry is quite satisfactory.

Figure 3: Channel with bump: Left: pressure, 20 isolines. Right: logarithm of the Mach number, 80 isolines. Although the Mach number exhibits a strong discontinuity, the pressure field is continuous.

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Finite volumes for complex applications

6. Bibliography

[Cl 99]

CLERC S., Accurate computation of contact discontinuities for flows with general equations of state, Computer Methods in Appl. Mech. Engrg., to appear.

[Cl 98]

CLERC S., On the preconditioning of finite volume schemes, 7th Int. Conf. on Hyperbolic Problems, Zurich, February 1998, Birkhauser.

[GuiVio]

GUILLARD H., C. VIOZAT, On the behavior of Upwind Schemes in the Low Mach number limit, Computers & Fluids, 28, 1999.

[PCV+]

PAILLERE H., CLERC S., VIOZAT C., TOUMI I., MAGNAUD J.-P., Numerical methods for low Mach number thermalhydraulic flows, Proceedings of the 4th ECCOMAS Conf. on Comput. Fluid Dynamics, Athens, 1998. John Wiley & sons.

[To 91]

TOUMI I., A weak formulation of Roe's approximate Riemann solver, J.C.P., 102, 1991, pp. 360-373.

[Tu 87]

TURKEL E., Preconditioned methods for solving the incompressible and low speed compressible equations, J.C.P., 72, 1987, pp. 277-298.

[ViMo]

VINOKUR M., MONTAGNE J.L., Generalized flux-vector splitting and Roe-average for an equilibrium real gas, J.C.P., 89, 1990, pp. 276-300.

Transient flows in natural valleys computed on topography-adapted mesh

S. Soares Frazao, J. Lau Man Wai, Y. Zech Universite catholique de Louvain Place du Levant 1 B - 1348 Louvain-la-Neuve

ABSTRACT The paper deals with the finite-volume computation of severe transient flows in natural valleys. The major problem concerning those flows is to have an appropriate representation and treatment of the topographical source term. A lateralization of the intercell flux is presented here to account for the bottom slope. Computation efficiency is gained by using a topography adapted mesh. Finally, an application to a Belgian valley is shown. Key Words: Finite-Volume, topography, source terms, mesh

1. Introduction Severe transient flows in natural valleys occur during extreme flood events, or as a result of a dam-break, which can be considered as the worst case. Several Finite-Volume numerical schemes have been developed for the computation of such flows, mainly focusing on an accurate evaluation of the numerical flux between cells. However, in natural topographies, important source terms appear due to the shape of the valley itself, they can also have relevant effects on the wave propagation. Therefore, an important step for a reliable computation is thus an accurate representation of the valley complexity, requiring a suitable mesh generation, in order to achieve a proper calculation of the topographical source term.

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Finite volumes for complex applications

2. Numerical scheme

A Finite-Volume numerical scheme is used to solve the Saint-Venant shallowwater equations. Those are written, in the 2D case,

where subscript t and x denote time and space derivatives respectively. The vectors of dependent variables £7, fluxes F(U) and G(U), and source terms S(U) are defined as

with h the water depth and it, v the depth-averaged velocity components. So and Sf are the bottom and friction slope, respectively written for the x- and y-direction. Friction slope Sf can be evaluated from the Chezy, Manning or another empirical formula. Integrating (1) on a discrete cell yields the following finite-volume scheme

where A is the cell area, Lj is the length of the cell interface j and n the number of cell interfaces. The vector U is the vector of rotated hydraulic variables obtained by applying the rotation matrix T to the original vector U and is aligned with the new co-ordinates axes (x , y ) where x is the direction normal to the considered cell boundary. The numerical fluxes between two cells located at the left and right side of the interface (cells / and r) are then calculated by the Boltzmann model [SOA 99]

Fields of application

405

Figure 1: Lateralization of the topographical source term where Fr is the Froude number. Fr represents the local Froude number, calculated with the velocity normal to the interface. 3. Lateralized topographical source terms

In each discrete cell, the bottom level zb is assumed constant. To compute the topographical source term S0X = — ^, the concept of flux lateralization [CAP 96] is used, and adapted to 2D problems. This means that the flux coming out of the cell located at the left side of the interface differs from the flux entering the cell located at the right side by a quantity gdz(hieft — ^-) accounting for the change in the bottom level (see figure 1, hatched area). The hydrostatic pressure term g^- in the flux is calculated with h being (hieft — dzb} or (hright) for the left and right cell respectively. This ensures that for water at rest, the pressure terms and the topographical source term, which are the only non-zero terms, cancel each other out. According to Nujic's compatibility condition [NUJ 95], the water will thus stay at rest. The lateralization is a very robust method, as no special reduction of the time step (or of the CFL number) is needed. However, it must be stated that it is probably not the most accurate method, especially for 1D problems. Indeed, in some cases, influence on the wave front velocity was observed, leading to a too slow flow propagation. 4. Topography-adapted mesh generation

The description of the valley topography coming from a DTM (Digital Ter-

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Finite volumes for complex applications

Figure 2: Definition of the segment density rain Modeling) usually consists of a square grid of points giving the bottom elevation. For dense grids, it is likely that the computation time becomes too important when considering all the available points. On the other hand, building a uniform and coarser mesh upon the given square grid can lead to accuracy losses and interpolation errors. In fact, rather flat areas could be represented with a lower mesh density than regions with abrupt changes in the topography and local steep slopes. A triangular mesh based on a given DTM can be constructed as follows. A first triangulation is generated only with the points defining the boundary of the computational domain. Additional intermediate points are then added one by one on existing segments, until the requested refinement has been reached. The mesh segments to be divided are selected according to their density. A segment between two points A and B (see figure 2) is said to have a high density when the error made by replacing the exact topography between A and B by a straight line is important. The error is quantified in the following way :

After each addition of a new point, the mesh is checked locally to avoid triangles with small internal angles. This leads to a local Delaunay mesh, with well conditioned triangles, and a finer discretization where needed, according to the given topography. 5. Application to the valley of Robertville The valley of Robertville is situated in the Eastern part of Belgium. A concrete gravity dam is located at the upstream end of the computed reach. The valley is steep and narrow, and widens at the downstream end, near the city of Malmedy (see figure 3). The aim of the study that was carried out was to determine the arrival time

Fields of application

407

Figure 3: valley of Robertville - distances in [m] of the water at the city of Malmedy in case of a dam break. A DTM was available for the valley, the grid size being of 40m. 5.1 Estimation of the friction

coefficient

A Manning friction coefficient of 0,02 sm- 1 / 3 was estimated for the valley. It is clear that a unique friction coefficient for the complete valley is not the most accurate choice, but this was due to the lack of information. A sensitivity analysis was thus carried out and showed the importance of this parameter on the wave propagation velocity. 5.2 Computed results The failure of the dam is considered as instantaneous. Several computations were run, and the criterion selected to compare the results is the arrival time at the city of Malmedy. A first run with a ID model gave a very short time (see Table 1). Computation were then run with different triangular mesh refinement levels identified by the maximum cell interface length : 100m, 70m 60m. The square grid given by the DTM was also used to compare with the other meshes. Figure 4 shows the inundated area once the water has reached the city of Malmedy. The city is located in the bottom left corner of the figure and the upstream reservoir is located at the right side of the dam. The sharp bends and the very narrow shape of the valley clearly appear. It can be seen in Table 1 that for triangular meshes, the travel time decreases

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Finite volumes for complex applications

Figure 4: Inundated area - distances in [m] with the mesh refinement. However, it is not expected to reach the short travel time of the ID model, even with further refinement. Indeed, the ID model considers the valley as being straight, i.e. without any bends, which is absolutely not the case for Robertville. The influence of bends on the flow was shown in [SOA 99]: important head losses occur, that slow down the wave front considerably. Sharp bends can even lead to partial reflection of the flow, with a bore travelling back in the upstream direction. The square grid appears to lead to the longest travel time. A possible explanation is the important error made on the flow path when the valley is not aligned with the mesh (see figure 5). A similar feature can be observed on mesh

time {minutes}

1D

6 17 21 31 50

2D triangulaaar (L = 60 m)

2D triangular (L = 70 m)

2D triangular (L = 100 m) 2D squaaree

Table 1: Arrival times at Malmedy

Fields of application

409

Figure 5: Influence of the square grid on the flow path triangular meshes, as the cell limits do not always match the exact wetted area delineation. However, with a triangular mesh, the error is less, as no preferential flow path is induced. Moreover, the fact that the mesh is generated according to the topography leads to a denser mesh at the steep river banks, allowing a more accurate representation of the wetted area delineation. 6. Conclusions

A lateralized treatment of the topographical source term of the 2D shallowwater equations was presented. Computations of a dam-break flow in a steep natural valley were shown on different kinds of meshes (triangular and square). The importance of an accurate representation of the topography of natural valleys clearly appears : the mesh used has a significant influence on the wave front velocity. Indeed, important differences in terms of wave propagation time were observed on the different runs. References

[SOA 99]

SOARES FRAZAO S., SILLEN X., ZECH Y., Dam-Break Flow Through Sharp Bends. Physical Model and 2D Boltzmann Model Validation , to appear in CAD AM, Proceedings of the first expert meeting, Wallingford UK

[CAP 96]

CAPART H., YOUNG D.-L., HUANG S.-Y., PAN J.-M., A lateralized flux-predictor scheme for a computation of openchannel flow in arbitrary topography , Proceedings of the 20th

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National Conference on Theoretical and Applied Mechanics, National Taiwan University, Taipei, China (1996) [NUJ 95]

NUJIC M., Efficient Implementation of Non-oscillatory Schemes for the Computation of Free-Surface Flows , Journal of Hydraulic Research , 33 (1), 1995

A mixed Finite Volume/Finite Element method applied to combustion in multiphase medium

Nathalie Glinsky-Olivier +, Eric Schall + + CERMICS-INRIA, Sophia Antipolis, France, + INRIA, Sophia Antipolis, France

ABSTRACT The present study is concerned with a mixed finite volume/finite element method applicable to triangular unstructured meshes for the simulation of wildfires propagating through a fuel bed using the multiphase approach. The forest fire is represented by a gas phase propagating through solid phases representing the heterogeneous combustible medium. Attention is focused on the development of a solution method of these equations. An explicit second-order in time and third-order in space accurate scheme has been constructed based on a special approximate Riemann solver applying the Roe's formulation. The radiative energy flux is calculated by the Discrete Ordinates Method. This model has been applied for simulating a wildfire through a litter of dead pine needles. Key Words: finite volume, finite element, multiphase approach, upwind scheme, combustion.

1. Introduction The present study is concerned with simulation of wildfires propagating through a fuel bed using a multiphase mathematical model. For this reason, attention was focused on an adaption of the Roe's scheme to our new set of equations. The source term includes many complex physical phenomena (heating, drying, pyrolysis of the fuel material and char combustion) and operates in each conservation equation.

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2. The physical model For this study, we have applied the multiphase formulation proposed by M. Larini et al. [LAR 99]. A brief description is given below. The fuel bed is supposed to be an heterogeneous medium made of solid particles of different types. A small control volume V contains N solid phases and a gaseous phase. Each solid phase is composed of particles of same thermochemical and geometrical properties (i.e. shape, size, arrangment) providing the same behavior under fire. The gas phase is a mixture of five species : CO, CO2, H2O, O2 and N2 numbered from 1 to 5 in this order. The packing ratio of the phase k is cx.k — Vk/V where Vk is the volume occupied by the phase k in the total volume V. In the same way, the fractional porosity is ag = Vg/V. We then have the relationship ag + X}fc=i ak — 1For the gaseous phase, we consider the conservation equations for the variable Wg where p — pg ag and pg is the density of the gas, (u, v] is the velocity of the gas, e is the specific internal energy and Yi are the mass fractions of the species with ^j=1 YI — 1. This relationship allows us to solve only 4 conservation equations for the species, the fifth mass fraction being deduced from the others. The set of conservation equations has been provided by M. Larini and B. Porterie [FOR 98] following the ideas in [GRI 85]. It can be written in conservative form :

where the notation Wt means dW/dt. The convective fluxes F and G write :

The pressure agp is calculated by using the perfect gas law for a gas mixture and the definition of the specific internal energy

where M; is the molar mass of the species i and Tg is the temperature of the gas. By combining these relationships, the pressure can be expressed as a function

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of 7, the ratio of the specific heats for the gas mixture

The diffusive fluxes R and S write:

where T> is the diffusion coefficient of the species. The stress tensors have the classical expressions

is the source term :

where riiH2o represents vaporization, mpr and msurf are the the solid mass loss due to pyrolysis and char combustion respectively. dragx and dragy are the two components of the drag force and g is the norm of the gravity. Qrad and Qcond represent the heat transfer between gas and solid phases due to radiation and conduction respectively. Ahi0 are the formations enthalpies of the species. More details can be found in [FOR 98]. The solid phase is described by the variable Wk [FOR 98] Wk ==• (ctkpk,Tk) where pk and Tk are respectively the density and the temperature of the solid phase A;. The mass balance and the heat conduction equations for the solid phase k write :

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where Cpk is the specific heat at constant pressure and Tfc is the temperature of the solid phase k. The solid phases are supposed to be at rest, then the momentum equations reduce to u^ = Vk — 0. These equations contain no flux, only a source term. The equations for the solid phase are solved separately from the gaseous phase equations both system being coupled by the source terms S19 and fi^ which depend on Wg and W&For the closure of the model, we distinguish two types of boundary conditions : the open sides (right, left and top of the domain) when a special upwinding is applied, and the lower boundary of the domain which corresponds to the ground where we apply no slip conditions for the velocity u = v = 0 and adiabatic temperature dT/dn = 0. The system formed with the convective fluxes is hyperbolic i.e. for any vector (771,772), the matrix M = r)idF/dWg -}-^dG / dWg is diagonalizable. Its eigenvalues are real and are

Then, we can write M = T-l(Wa,fl}\(Wg,iif}T(Wg,jf) where A = diag(\i] is the diagonal matrix composed of the eigenvalues A; of M and T is the invertible transformation matrix. The absolute value of M. writes \M\ — T~1(Wg,ff) \A.(Wg,ff)\T(Wg,f)). We can notice that the two last eigenvalues Ay and AS have different expressions from the classical reactive Navier-Stokes system. Their values are close to the usual eigenvalues u TJI + vr\2 ± \/7 p/p.

3. Spatial approximation We give here a short description of the numerical model, details can be found in [SCH 99]. Starting with a (possibly unstructured) Finite Element triangulation of the calculation domain D, dual Finite Volume cells are constructed joining successively the gravity centers of the triangles having a node i as a vertex and the middle of the sides adjacent to this node. We write the variational formulation of the gaseous system. The mixed FV/FE formulation traduces by the choice of the test functions \l>;. For the temporal, convective and source terms, the test function \I>; is the characteristic function of the cell Ci and for the diffusive part, \&; is the PI Galerkin basis function at the node i whose support is Si = UT,iETT . After integration of the different terms on their respective support, application of the Green formula and mass-lumping

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for the temporal and source terms, we obtain

rri

where Ai is the area of the cell Cj, (n^n^} is the outward normal to the cell Ci and (nx,ny) is the outward normal to the boundary of the calculation domain D. A first-order upwind approximation of the convective fluxes write

where % is the integral normal to the interface between CT and Cj and is the numerical flux through this interface. This numerical flux is calculated by applying a Roe scheme [ROE 81] adapted to our special convective fluxes F and G. For the calculation of the fluxes of the mass fractions equations, we use the multi-componenents flux proposed by B. Larrouturou [BLA 89], which allows to preserve positivity for the mass fractions. IfOPijis the numerical flux for the density variable, the flux of the species Y is :

High-order spatial accuracy is obtained via the M.U.S.C.L. method [VLE 72], [LFE 85] and a /3-scheme formulation which consists in writing the numerical flux $ (Wij.Wj^ffij) with for instance W%j = Wi + 1/2[(1 - 2/3)(W3 - WJ + 2(3VWi.ij where (3 is an upwinding parameter. Choosing (3=1/3 provides a third-order accurate scheme in space. A PI Finite Element interpolation for the approximation of the diffusive fluxes leads to

where AT is the area of the triangle and R\T and S\T are constant on T. More details can be found in [LFE 89]. The multiphase radiative transfer equation is solved using the Discrete Ordinates Method [SAK 96]. Since the soots are not included in this model, the contribution of the radiation is negligible compared to the other physical phenomena. A detailed description of this implementation can be found in [SCH 99].

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3. Numerical results This numerical model has been applied for describing the behaviour of a spreading wildfire through a litter of dead pine needles. The domain is 1m height and 2m long and the fuel depth is 0.04m. The mesh is composed of 5151 nodes. Test conditions are the same as described in [POR 98]. The fire is ignited at the middle of the litter and the ignition temperature of the solid is equal to TOOK since the gas is at ambient temperature. We present the isovalues of the gas temperature, of H2O and CO2 at two different times (0.26s and 1.08s) showing the evolution of the phenomenon. At time 1, an energy convective-conductive transfer is well observed. We also notice the presence of water vapor (H-2Omax = 0.14) and the formation of pyrolysis gases (CO2max = 0.28) while the gas temperature reaches Tmax = 1717.K) Later, at time 2, an increase in gas temperature (Tmax = 3068.K) is due to the combustion of the pyrolysis gases (CO-2max = 0.095) and the decrease of the water (H2Omax = 0.017). The propagation to the left of the flame front can be observed : the maximum values of the variables have moved towards the unburnt part of the litter. 4. Conclusions A mixed Finite Volume/Finite Element method to solve a 2D multiphase radiative and reactive model of line-fire propagation is proposed. The different physical phenomena are well observed (heating, pyrolysis and combustion). The first results obtained with this model are very encouraging. Physically, a model including soot formation will be considered to show the importance of the radiation in such combustion problems. Numerically, this kind of unsteady problems is very time consuming. An Adapted accurate implicit solver is in progress. Aknowledgment The authors thank M. Larini and B. Porterie for having provided the physical model and the set of equations. This work has been financed by the European Economic Commission under the contract EFAISTOS No.ENV4-CT96 0299. 5. Bibliography [LFE 85]

FEZOUI, L., Resolution des equations d'Euler par un schema de van Leer en elements finis, Rapport de Recherche INRIA No.358, Janvier 1985.

[LFE 89]

FEZOUI L. et al., Resolution numerique des equation de Navier-Stokes pour un fluide compressible en maillage triangulaire, Rapport de recherche INRIA No. 1033, mai 1989.

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[GRI 85]

GRISHIN A.M. et al., Study of the structure and limits of propagation of the front of an upstream forest fire, translated from : Fizika Goreniya i Vzryva, 21(1), pp.11-21, 1985.

[LAR 99]

LARINI M. et al., A multiphase formulation for fire propagation in heterogeneous combustible media, Int. J., of Heat and Mass Transfer (to be published).

[BLA 89]

LARROUTUROU, B. et al, On the equations of multicomponent perfect or real inviscid flow, Carasso, Charrier, Hanouzet et Joly Editors, Non linear hyperbolic problems, pp.69-98, Springer-Verlag Heidelberg, 1989.

[FOR 98]

PORTERIE B. et al., Wildfire propagation : a two-dimensional multiphase approach, Combustion, Explosion and Shock waves, Vol.34, No.2, 1998 also in Fizika Goreniya i Vzryva, Vol.34, No.2, 1998.

[ROE 81]

ROE, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J.C.P., 43, pp.357, 1981.

[SAK 96]

SAKAMI M. et al., Application de la Methode des Ordonnees Discretes au Transfert Radiatif dans un Milieu Bidimensionnel Gris a Geometrie Complexe, Rev. Gen. Therm., Vol.35, pp.83s-94s, 1996.

[SCH 99]

SCHALL E. et al., A mixed Finite Volume/Finite Element method applied to combustion in multiphase medium, INRIA Report, to appear.

[VLE 72]

VAN LEER B., Towards the ultimate conservative difference scheme I: the quest of monotonicity, Lecture notes in Physics, Vol.18, pp.163, 1972.

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Figure 1: Isovalues of the temperature - time 1 and time 2

Figure 2: Isovalues of H2O - time 1 and time 2

Figure 3: Isovalues of CO? - time 1 and time 2

Turbulence Modeling for Separated Flows

L.J. Lenke and H. Simon Department of Turbomachinery, University of Duisburg D-47048 Duisburg

The influence of the turbulence modeling on viscous flow field calculations has often been discussed in the past. For a meaningful comparison of different turbulence models the access to reliable measurement data is necessary. Therefore, the 2-D flow through a symmetrical diffuser with a large asymmetrical separation is chosen to investigate the separation behavior of three different two-equation turbulence models. The k-uj-, a low-Reynolds-number k-e- and an explicit algebraic Reynolds stress model are considered in this investigation. The numerical results of the algebraic Reynolds stress model are similar to the k-cj-model in prediction of the reattachment length but are in better agreement with experimental data downstream the reattachment. Significantly improved numerical results were obtained compared to the k-e-model. Furthermore, different 3-D calculations of the flow within the return channel of a multi-stage centrifugal compressor are presented. Especially at off-design conditions the turbulence models deviate notably in their prediction of separation and show great differences in the calculation of the 3-D flow structure and secondary flows. Key Words: k-cj, k-e, algebraic Reynolds stress model, separation, return channel

diffuser,

1. Introduction There are already a large number of two-equation turbulence models but a basic problem of these kind of models, namely, their failure to correctly predict the amount of separation in adverse pressure gradient flows, is still unresolved. Among the various models in existence, the fc-e-model is currently most popular and applicable to many practical complex flows with reasonable computational

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economy and accuracy. However, for the prediction of large flow separations and recirculations especially with regard to the prediction of the flow downstream of the separation the k-e- and fc-w-model have to be abandoned in favor of the higher order turbulence models which will increase strongly the numerical effort. Another point of criticism is that the fc-e-models need damping functions within the boundary layer. Recently, the explicit algebraic Reynolds stress models (ARS) which are applied in the present paper within the context of the fc-e-equation extend the range of applicability of two-equation models. This will be illustrated by two test cases (2-D diffuser and return channel) involving separations at plane and curved surfaces. The 2-D flow through the diffuser is characterized by a large asymmetrical separation at one side of the diffuser. The second test case show the 3-D flow within a return channel which is typically to join the exit from one stage of a centrifugal machine to the inlet of the next stage. 2. The Numerical Scheme

In this investigation the code developed by Reichert [REI 95] with a finite volume formulation of the full Navier-Stokes equations is used picking up elements from Roe's and Osher's scheme. Furthermore, only steady state solutions are considered. For high convergence rates, an implicit Newton-Raphson-like iterative method is used to solve first the averaged conservation equations and then the modeled transport equations but with different CFL numbers for each set of equations. Furthermore, the local time step size is calculated using a CFL number, which is a function of the local change of the density (details are described in Lenke and Simon [LEN 97a]). For the simulation of turbulent flows non linear eddy-viscosity models are an increasingly popular approach, motivated principally by the desire to combine the physical realism offered by second-moment closure with the simplicity and numerical robustness of linear eddy-viscosity models. Gatski and Speziale [GAT 93] derived an explicit algebraic stress equation for three-dimensional turbulent flows which must be solved in conjunction with two transport equations. In the present study the algebraic stress equation is solved with the standard fc-equation and an e-equation which is extended by an additional production range time scale and a cross-diffusion term (ARS-model) to improve the separation behavior of the turbulence model (details are described in Lenke and Simon [LEN 97b]). This ARS-model will be compared with the low-Reynolds-number fc-e-model devised by Lam and Bremhorst [LAM 81] and the fc-u-model devised by Wilcox [WIL 91]. 3. Boundary Conditions

At the inlet boundary, in both test cases the averaged total pressure and

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total density were specified. Furthermore, the distributions of the velocity components were taken from the measurements. At the outlet, in both cases a variation of a constant static pressure was used to establish the correct mass flow. All solid surfaces were modeled as rigid, non-slip and adiabatic. The other flow quantities were extrapolated from the interior. In both cases no values about the measured turbulent kinetic energy at the inlet are available so that k and e were extrapolated from the interior which leads to an averaged turbulence intensity of nearly 1%.

4. Numerical Results

4.1. 2-D

Diffuser

The performance of the two-equation turbulence models were evaluated for the flow through a 2-D symmetrical diffuser with a generating angle of 20 = 14.25°. A 118x65 grid was used and the numerical results were compared with measurements taken from Gersten et al. [GER 87]. The flow through the diffuser is characterized by a large asymmetrical separation at one side of the diffuser which is shown in Fig. la. All the turbulence models calculated such a separation but there are great differences in the prediction of the reattachment point or velocity profiles. Fig. Ib shows the velocity distributions near the beginning of the diffuser. At this position the ARS- and fc-u;-model calculate a very small backward flow which is similar to the measurements. The k-emodel calculates a symmetric velocity distribution which shows no similarity with the measurements. Fig. Ic and Id show the velocity distributions within the separation and demonstrate the underprediction of flow separation with the fc-e-model in this case. The two other models are similar and differ only within the separation. At this location the ARS-model calculates the highest negative velocities and agrees very well with the measurements. Near the reattachment and especially downstream the reattachment the results demonstrate the superiority of the ARS-model in relation to the standard two-equation models. The fc-w-model overpredicts the reattachment length and calculates downstream the reattachment great differences between the measured and predicted velocity distribution (Fig. Id and e). The ARS-model calculates the reattachment nearly correct with the highest negative velocities within the separation and the highest positive velocities downstream of the separation. 4.2. Return Channel The calculations to be presented have been done for the flow through a return channel with a small flow coefficient and a design flow angle 03 — 26° at the inlet. The geometry of the channel and the computational H-grid are

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shown in Fig. 2. The grid is a single block grid with y+ < 1. Only near the leading edge y+ has maximum values of 5 in a very small region. Two streamlines illustrate the flow at design point. The geometry of the channel is characterized by a constant channel width between point 3 and 4 with an increasing cross-sectional area up to point 4' and a decreasing cross-sectional area up to point 6. The Mach number distribution within the channel at design point is shown in Fig. 3. Due to the high circumferential component of the velocity at the inlet and the small channel width, the velocity distribution at the exit of the 180°-bend is uniform between hub and shroud. The secondary flow has no great influence on the flow. The differences between the turbulence models are small so Fig. 3 show only the Mach number distribution of the ARS-model. Each turbulence model calculate a large separation at the suction side which increases from hub to shroud and reattaches in the rear half of the vane. The models differ only by the prediction of the separation length with the smallest separation calculated by the ARS-model. This behavior is in contrast to the results of the 2-D diffuser which demonstrated the fact that the fc-e-model usually underestimates large separations. With higher mass flow rates (0:3 = 33°) the differences between the turbulence models increase. Due to the incidence angle the ARS- and fc-w-model calculate a separation near the leading edge at pressure side (Fig. 4). The k-cmodel calculates this separation too small with a higher pressure distribution in this region (Fig. 6). In the middle of the suction side the separation behavior of the turbulence models changes. The ARS-model calculates at suction side a smoother pressure distribution which agrees better with the measurements. This results by a smaller separation at mid span compared to the k-eand fc-cu-model. In contrast to the ARS-model the fc-cj-model calculates always larger separations compared to the fc-e-model which agree better with the measurements at pressure but not at suction side. Within the 180°-bend and in the range of the leading edge the Mach number distributions of the turbulence models are very similar. But downstream of the separations the ARS-model calculates relatively high values of the turbulent viscosity within the whole channel. This leads to a very homogeneous Mach number distribution between the rear half of the vanes and within the 90°bend. The k-e- and fc-w-model calculate in wide areas no turbulent viscosity like a laminar flow with more vorticity especially within the 90°-bend (Fig. 5). But the great differences between the Mach number distributions have no significant influence on the static pressure distribution. 5. Conclusions An explicit algebraic Reynolds stress model has been tested against a low Reynolds number k-e- and fc-w-model for a diffuser flow which involves a large separation. The comparisons with experimental data show that the ARS-model

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does provide significant improvements over the standard two-equation models especially with regard to prediction of flow separations. For the flow through a return channel which is characterized by strong stream line curvature the comparison of the turbulence models shows minor improvements of the ARSmodel. These improvements mean not only the enlargement of separation at pressure side like the k-u-mode\ but also the reduction of separation at suction side. Furthermore, the ARS-formulation used in the framework of fc-e-equations is as robust as fc-e-models. Thus, these kind of turbulence models increase the range of applicability of two-equation turbulent closure formulation at a minimal increase in cost and storage. 6. Bibliography [GER 87]

Gersten, K., Hartl, A., Pagendarm, H.G., Optimierung von Diffusoren bezuglich der Diffusorstromung und der Diffusorwande. VDI-Verlag Dtisseldorf 1987.

[GAT 93]

Gatski, T.B., Speziale, C.G., On explicit algebraic stress models for complex turbulent flows. Journal of Fluid Mechanics 254 (1993) 59-78.

[LAM 81]

Lam, C.K.G., Bremhorst, K.A., Modified Form of the ke-Model for Predicting Wall Turbulence. Journal of Fluids Engineering 103 (1981) 456-460.

[LEN 97a]

Lenke, L. J., Simon. H., Viscous Flow Field Computations for a Transonic Axial-Flow Compressor Blade Using Different Turbulence Models. ASME Paper 97-GT-207 (1997).

[LEN 97b]

Lenke, L. J., Simon. H., An Improved Algebraic Reynolds Stress Model for Predicting Separated Flows. Beijing, Proceedings of the 7th International Symposium on Computational Fluid Dynamics, 15.-19. September 1997.

[REI 95]

Reichert, A.W., Stromungssimulationen zur optimierten Gestaltung von Turbomaschinenkomponenten. Dissertation, Duisburg, Hansel-Hohenhausen Verlag 1995.

[ROT 84]

Rothstein, E., Experimented und theoretische Untersuchung der Stromungsvorgange in Rilckfuhrkanalen von Radialverdichterstufen, insbesondere solcher mit +geringen Kanalbreiten. Dissertation, Aachen 1984.

[WIL 91]

Wilcox, D.C., Comparison of Two-Equation Turbulence Models for Boundary Layers with Pressure Gradient. AIAA Journal 31 (1993) 1414-1421.

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Figure 1: Flow through a 2-D diffuser with 20 = 14.25° (a) streamlines with the ARS-model; b)-e) velocity distributions)

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Figure 2: Streamlines and grid of the return channel (44^-37x162 points)

Figure 3: Mach number distribution at design point (0:3 = 26°).

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Figure 4: Velocity and Mach number distributions within the return channel (ARS-model; a3 = 33°).

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Figure 5: Mach number distributions within the return channel (k-€-model; a3 - 33°j.

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Figure 6: Surface pressure distributions at mid span (a$ = 33°).

Simulation of unsteady Flow in a Vortex-Shedding Flowmeter

Stephan Perpeet, Andreas Zachcial and Ernst von Lavante Institute of Turbomachinery, University of Essen, D-45127 Essen, Germany

ABSTRACT Most commercial vortex-shedding flowmeters rely on known relationship between the vortex-shedding frequency and the mass flow, needing regular and well defined vortex structure as well as shedding mechanism. However, it has been observed that some designs result in rather irregular pressure signature of the vortex system, leading to problems in the signal processing. In the present study, the flow about the bluff body in a vortex-shedding flowmeter was numerically investigated using a solver of the unsteady, compressible Navier-Stokes equations in two and three dimensions. The computations were compared with experimental results obtained by ultrasonic measurements downstream of the bluff body. Several different body shapes were studied, trying to optimize the resulting pressure signature downstream of the body. Furthermore, first results of the influence of pulsations on the flow will be presented. Key Words: Vortex-shedding flowmeter, Low Mach number, Pulsation

1. Introduction Many aerodynamic problems require a volume- or mass-flow data for its quantitative solution. Therefore, a number of methods for flow rate measurement have been developed. In the present work, attention was paid to the so-called vortex-shedding flowmeters. Commercial flowmeters use a large variety of bluff body shapes, often restricted by the attachment of the pressure sensors or, more likely, the patent laws. In principle, the vortex-shedding flowmeters use the separation frequency of vortices behind a bluff body to measure the mean flow velocity of a fluid flow (Figure 1). Downstream of the bluff bo-

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dy, a von Karman vortex street develops; it's width and distance between the vortices depends on this frequency, and therefore on the bluff body's shape. Preferably, the vortex-shedding frequency should depend linearly on the mean flow velocity for a wide Reynolds number range. The dependency of the vortex frequency /, the mean flow velocity um and the width of the bluff body d is expressed by the dimensionless Strouhal number:

Figure 1. Principle of a vortex-shedding flowmeter Previous simulations of some current bluff body designs lead to fairly irregular pressure signatures, making them unreliable. It was, therefore, decided to investigate a few alternate bluff body designs. In the presently used ultrasonic method, the bluff body shape is restricted only by the required minimum strength and rigidity to avoid vibrations. The design could be optimized regarding the pressure signature or the pressure drop across the body. The signal processing requires well-defined vortices at only one dominant frequency, without any secondary effects. Therefore, an economic signal processing is only feasible with an optimized shape of the vortex body. 2. Numerical Algorithm

The numerical algorithm employed uses the three-dimensional, time-dependent full Navier-Stokes equations describing the conservation of mass, momentum and energy of the flow. The divergence form in body-fitted, curvilinear coordinates is:

with Q — J~l(p pu pv pw e)T the vector of the conserved variables. J is the Jacobian of the coordinates transformation from physical ( x , y , z , t ) to computational (£,r7,£,r) space. The program is based on the finite-volume formulation, using a cell-centered organization of the control-volumes. The spatial discretization is carried out with the help of Roe's Flux Difference Scheme,

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a Godunov-type method providing an approximate solution of the Riemann problem on the cell interfaces. Here, the flux is [VAT 87]:

The index i + | describes the values on the cell interfaces, R and L the values right and left from it. The Roe-averaged matrix A is given by the differentiation of the local-linearized function F(Q). This scheme is formally a central difference type plus a damping term. The method has been proved to be very accurate and effective in the simulation of low Mach number viscous flows [LAV 93]. Upwind-biased differences are used for the convective terms, central differences for the viscous fluxes. Starting with a constant initialization of the scalar variables and body-fitted velocity components, the integration in time is carried out by a modified explicit Runge-Kutta time stepping as well as, optionally, an implicit Approximate-Factorization method (AF) or SymmetricGauss-Seidel (SGS) scheme. 2.1. Low Mach Number Modifications Numerical algorithms for the simulation of compressible flows become inefficient and inaccurate at very low Mach numbers. The difficulties are due to the formulation of the governing equations in their discretized form. The problems have been addressed by many previous investigators, including Shuen et al. [SHU 93], Fletcher et al. [CHE 93] and Edwards et al. [ROY 98]. The main problem is the stiffness of the governing equations at low Mach numbers. The condition number of the Jacobian matrices (ratio of the maximal to the minimal eigenvalues) increases to infinity as the Mach number approaches zero. The time marching step therefore is restricted because of the large disparity of the eigenvalues of the Jacobian, representing the convective and acoustic signal speeds, respectively. The second problem results from the pressure singularity in the momentum equations at very low Mach numbers. The ratio of the magnitudes among the pressure term p and the convective terms pu2 is inversely proportional to the Mach number squared M2. The large difference in magnitude will yield a large roundoff error. To enable efficient numerical solutions of the equation system at low Mach number, a pseudo-time term is added to the time-dependent compressible Navier-Stokes equations. The primitive variables are employed as unknowns, rather than the traditional conservative variables. A preconditioning matrix F is used in order to eliminate the time-step difference between the convective and acoustic characteristic speeds at low Mach number. The preconditioned

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equations can be discretized in the delta form:

where the Jacobian matrices are defined as,

Here Q is the vector of the conservative variables and Q is the vector of the primitive variables (p, u, v,w, T)T. A more detailed description of the matrix F can be found in [YAO 97]. To circumvent the problem of large difference in magnitude of the convective and pressure terms in the momentum equations, a gradient splitting of the Euler flux into convective terms and a pressure term can be considered: Both split gradients are of the same order of magnitude, independent of the Mach number. The Liou's Advection Upwind Splitting Method is applied to treat the convective and pressure terms separately. The convective terms are upstream-biased using an appropriately defined advection Mach number at the cell interface, while the pressure term is strictly dealt with by using acoustic waves. One of the advantages of the method is that the upwind effect can be easily reached with few modifications of the programs. 2.2. Verification The present numerical algorithm was subjected to verification of it's temporal and spatial accuracy and consistancy. The scheme is formally second order accurate in space, since the viscous terms are obtained from second order central differences. The scheme was first verified using the usual grid refinement study for the case of viscous flat plate flow at a free stream Mach number of MOO = 0.5. Defining the global error as the I/2 - norm of the deviation of the present solution from the Blasius solution, second order accuracy was verified (von Lavante [LAV 90]). Next, the combination of the present solution scheme with the computational grid was investigated for the case of the "new design" bluff body (see below) using three different grids. Table 1 summarizes the results. It should be noted that the finest grid very closely approaches the Strouhal number of the corresponding experiment.

Fields of application Meshpoints

A* [a] rel. CPU-time

f(Hz] Strouhal number

5000 8.8 • 10-7 1 342 0.267

12300 5.88 • 10~7 10 339 0.265

433

41600 || exp. 7 2.06 -KT 35 335 0.262 0.263

Table 1. Comparison of different grids The temporal accuracy was tested by simulating the 2-D flow about a cylinder, with free stream Mach number M^ = 0.1 and a Reynolds number of Re = 200. The resulting Strouhal number of the vortex separation is compared with known experimental and numerical results in Table 2. The agreement is also very good. Lugt Truckenbrodt Authors Faden

Method experimental experimental numerical numerical

Strouhal num. 0.193 0.192 0.194 0.198

Table 2. Strouhal number in cylinder wake flow 3. Results In the present research project, several shapes of the bluff body were investigated experimentally and numerically. The experimental results have been given in detail by Hans et. al. [HAN 98]. Main emphasis was put on linear dependency of the vortex-shedding frequency as a function of the mean flow velocity. In the present paper, only the three most interesting bluff body shapes, shown in Figure 2, will be discussed. Two of the bodies, a "T"-bar and a rectangle, were fairly simple, the third one was optimized by the present authors for the cleanest signal and was therefore called "new design".

Figure 2. Design of the bluff body shapes The horizontal part of the T-bar was facing upstream, which was in contrast to other applications of this shape [MIA 93]. The Strouhal number (Figure 3) was constant for the whole velocity range corresponding to Reynolds numbers

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from 10,000 to 300,000. Experimental and numerical results showed a very good agreement. The standard deviation was within 2%.

Figure 3. Strouhal number versus Reynolds number Although the two-dimensional simulations predicted vortex-shedding frequencies that agreed very well with the experimental data, it was decided to investigate the flow about the promissing "new design" using three dimensional approach. No symmetry was assumed, allowing the specification of asymmetric inflow velocity profiles. The dominant vortex structure makes this case interesting for large eddy simulation (LES) turbulence treatment. Therefore, no turbulence modell was used, although the inflow would correspond to a fully developed turbulent pipe flow. The computational grid consisted of 22 blocks. Two grid resolutions were used: coarse with 300, 000 cells and fine with 1.07 • 106 cells. The coarse grid was implemented in order to study the process of the main vortex generation. No attempt was made in this case to properly resolve the boundary layer at the pipe wall. There were only a few grid points in this boundary layer. The second, finer grid consisted of 1.07 • 106 points, making the full simulation of all the viscous layers possible. The minimum grid spacing at the wall was less than y+ = 1.0, and of the same order of magnitude as the Kolmogorov length. There were at least 20-40 grid points in each of the shear layers. The grid was exponentially stretched toward regions with smaller gradients of the flow variables. The resulting flow displayed highly three-dimensional nature, with main vortices curved toward the walls, and with a complex system of secondary flow features. After separating from the bluff body, the primary vortices, seen in Figure 4, were convected downstream. They dissipated toward the pipe walls; their central part became straight, assuming almost two-dimensional form. In order to visualize some of the lower order features, such as secondary vortex structure, the enstrophy, defined as uj — \ (V x U)2 was computed and displayed in contour plots. The enstrophy contours on a cylindrical surface close to the pipe wall are shown in Figure 5. Here, the so called "horse shoe vortex" is clearly visible in front of the bluff

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body. Behind it, the highly vortical, unsteady flow associated with the primary vortices can be seen. Several secondary and tertiary vortices, some aligned with direction normal to the main vortices, could be observed, indicating that at least some of the turbulent structure was simulated.

Figure 4. Density contours over the pipe cross-section

Figure 5. Enstrophy contours at the wall

Figure 6. Pressure evolution for pulsation frequency of lOOHz

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Finally, the effects of pulsating inflow on the function of the vortex-shedding flowmeter are currently investigated. As can be seen in Figure 6, pulsations in the flow with frequencies close to the natural vortex-shedding frequency had an pronounced influence on the operation of the meter. Between two pulsation peaks the former shedding frequency recovers. 4. Conclusions

A verified and validated Navier-Stokes solver for two-dimensional and threedimensional simulations of compressible, viscous and unsteady flow over a wide range of Mach numbers is presented. The solutions achieved by this program show good agreement with experimental data. The effects of disturbed inflow conditions are currently investigated. 5. Acknowledgement

This project is supported by the Deutsche Forschungsgemeinschaft (DFG).

6. References [ROY 98] [HAN 98]

[MIA 93]

[CHE 93] [SHU 93] [VAT 87]

[LAV 90] [LAV 93]

[YAO 97]

J. R. EDWARDS AND C. J. ROY, AIAA Journal, Vol. 36, No. 2, pp. 185-192, 1998. HANS, V., POPPEN, G., LAVANTE, E. v., PERPEET, S.: Vortexshedding flowmeters and ultrasound detection: signal processing and bluff body geometry. Flow Measurement and Instrumentation 9 (1998), 79-82. MIAU, J.J., YANG, C.C., CHOU, J.H. AND LEE, K.R., A T-shaped vortex shedder for a vortex flowmeter, Flow Meas. and Instr., Vol. 4, No. 4, 259 - 267, 1993. R. H. PLETCHER AND K.-H. CHEN, AIAA-93-3368-CP. J.-S. SHUEN, K.-H. CHEN, AND Y. CHOI, J. of Comp. Phy. 106, pp. 306-318, 1993. VATSA,V.N.,THOMAS,J.L.,WEDAN,B.W., Navier-Stokes computations of prolate spheroids at angle of attack, AIAA Paper, (87-2627), 1987. VON LAVANTE, E., The Accuracy of Upwind Schemes Applied to the Navier-Stokes Equations, AIAA Journal, Vol. 28, No. 7, 1990. VON LAVANTE, E., YAO, J., 1993. Simulation of flow in exhaust manifold of an reciprocating engine. AIAA 24th Fluid Dynamics Conference. J. YAO AND E. VON LAVANTE, Proceedings of 7th International Symposium on CFD, pp. 689-694, Sept. 15-19, 1997, Beijing, PR China.

A Finite Volume Scheme for the Two-Scale Mathematical Modelling of TiC Ignition Process

A. Aoufi LIMHP-CNRSs Avenue JB Clement 93430 Villetaneuse

V. Rosenband Faculty of Aerospace Engineering Techmon Haifa-32000

ABSTRACT We consider the problem of the ignition of a cylindrical sample of infinite length composed of Ti particles coated with a thin TiC layer and placed into a furnace at T. We describe an implicit finite-volume scheme, for the two spatial scales modelling, on a moving mesh at the particle level, applied to the numerical study of the ignition process. Key Words: Reaction-Diffusion Equation, Adaptive Finite- Volume, Stefan Problem, Moving-Mesh, Ignition Process, Solid-Solid Combustion.

1. Introduction

In this paper we present a two-spatial scales mathematical modelling for Ti + C ->• TiC ignition process [AOU 98, LAP 96, SEP 95]. At the particle level, we have a Stefan problem which is discretized by a finite-volume scheme on a moving mesh. The paper will be organized as follows, in section 2 we describe the mathematical modelling, in section 3 we present the numerical scheme we have applied to the treatment of this two-scale , and describe the finite volume scheme. Section 4 presents some numerical simulations.

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2. Mathematical Modelling The reader is referred to [AOU 98] for more explanations about the physical analysis of the process.

Figure 1: Scheme of the reaction modelling

2.1 Mass Balance We consider the mass-diffusion balance expressing Pick's law for a spherical particule of Titanium of radius Rp. We denote by Q = f^2 (<)UQi (t] = [Q,R(t)] U [R (t) , Rp] = [0, Rp] .The mass-diffusion coefficients follow an Arrhenius-type law. 2.2.1 Dissolution Domain , Zone II

2.2.2 Carbidization Domain , Zone I

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2.2 Stefan Problem We express the condition at the interface which indicates that due to concentration gradient in both domains I and II, the interface moves with the velocity

2.3 Energy Balance We evaluate the overall energy balance at the sample macroscopic level, for which we take into account the heat released at the mesoscopic scale - i.e. particle scale- by the exothermic mass diffusion processes described below. 2.3.1 Carbidization Domain , Zone I We denote by j\ ' the rate of heat released during carbon formation - by diffusion process in zone I-, denned by

2.3.2 Dissolution Domain , Zone II 11

We denote by dO^dt\

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2.3.3 Global Heat Balance We consider the cylinder domain £lc = [0, RC] and write for V z G [0, RC] The heat balance takes into account the heat released locally at the particle level for the two diffusion processes by:

2.3.4 Boundary Conditions Symmetry Condition at the center of the cylinder

Heat supplied by the furnace maintained at temperature T^

3. Numerical Scheme

We integrate from tn to tn+\, the three different balances, assuming that a non uniform temporal mesh is given. We use a variable stepsize backward difference formula of order 1, which is well suited for the numerical treatment of stiff equations. We focuss on the numerical methodology used to discretise the mass balances, written in conservation form, on a variable domain. In order to track efficiently the variation of the computational domains QI (t) and ^2 (t) , we have considered the integration of the mass balance on a cell K i ( t ] = [ n ( 0 > r » + i ( 0 ] wmch evolves with time and denotes its boundary dKi (t) = {ri (t) , r,-+i (t)} . We adopt a cell-centered formulation. 3.1 Integration on a moving mesh

The transport theorem of Reynolds [N'KO 94] affirms that:

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where t>;+1 (t) = r 'dt ' and i;,- (t) = r^ ' are the velocities of the extremities of the intervall on which the mass balance is integrated. We outline the procedure, and omit the details of the algebraic computations. We integrate the previous relation from tn to tn+\ and substitute ^ ( z , r , t ) from its definition into the mass balance. We apply Green's Formula for the discretization of the divergence of the mass fluxes. We apply the first order accurate backward Euler scheme for the discretization of the right hand side integrals. We denote Atn = volume of the cell and end up with,

We point out that the following term, which takes into account the transport of the concentration field, due to the variation of volume of the integration cell appears :

Since C is constant in cell KI (tn) , we may use "local information" i.e. related to this cell, and therefore write the following continuity relation

We now have the following relationship

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After some algebra and leaving the convected mesh terms in the right hand side, we are led to a three diagonally dominant matrix system which can be efficiently inverted by the classical -double sweep- Thomas Algorithm. At this stage we have made no assumption on the nodes velocities. 3.2 Nodes Velocity We have to prescribe the speed of the mesh; this is realized through a linear speed formula, according to [MUR 59], taking into account the fact that the velocity at the interface between domain I and II is known from the stefan problem equation. 3.3 Boundary Conditions An important issue is the discretization of the boundary conditions on a moving mesh, because they will lead to a different set of equations that will influence significantly the behaviour of the numerical solution, if a wrong formula is supplied, for example consider: Constant We integrate the equation on cell Kj (tn) = assuming that therefore and the equation is,

Using the previous arguments on the continuity of the concentration field we get

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In fact the variation of the computational volume can be understood as a source term. 3.4 Time Step Adaptation The stiffness of the problem means that once the heat released by the chemical kinetics is enough, a sharp rise of the temperature occurs inside the sample from the exterior surface to the interior of the cylinder. A time-stepping adaption procedure to dynamically reduce the time-step with respect to the temperature field behaviour has been considered. 5. Numerical simulations

Ignition temperature is reached, when a sharp increase in temperature field occurs. This ignition process depends on particle size, initial thickness of TiC interfacial layer, heating rate, and value of TOO. We present below a sample of such temperature profile.

Figure 2: Temperature profile T(Rc,t)

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[AOU 98] . AOUFI A., ROSENBAND V., Mathematical modelling and numerical simulation of Titanium/Carbon Ignition , 2nd International High-Energy Materials Conference and Exhibit, Dec 8-10 (1998). [LAP 96]

. LAPSHIN O.V. AND OVCHARENKO A.E. , A mathematical model of high-temperature synthesis of the intermetallic compound Ni^Al during ignition , Combustion, Explosion and Shick Waves , Vol. 32, N° 2,(1996),158-164.

[MUR 59] MURRAY W.D. AND LANDIS F., Numerical and Machine Solutions of Transient Heat-Conduction Problems Involving Melting or Freezing , Transactions of the ASME, (May 1959), 106-112. [N'KO 94] N'KoNGA B. AND GUILLARD H., Godounovtype methods on nonstructured meshes for three-dimensional moving boundary problems , Comput. Meths. Appl. Mech. Engrg. 113 (1994). [SEP 95]

SEPLYARSKII B.S. AND GORDOPOLOVA I.S., Ignition of condensed systems interacting through a layer of high-melting products , Combustion, Explosion and Shock-Waves, Vol. 31, N°4, (1995), 405-410.

Two Perturbation Methods to Upwind the Jacobian Matrix of Two-Fluid Flow Models.

Kumbaro A., Toumi I. CEA Saclay, DRN/DMT/SYSCO F-91191 Gif-sur-Yvette Cedex, France [email protected] Cortes J. CEA Cadarache, DRN/DTP/SMET F-13108 Saint-Paul lez Durance, France

ABSTRACT We examine the eigenstructure of a two-fluid model Because of complex interphase interactions, perturbation methods are used to get approximations of the eigenelements. We compare two significant perturbation methods. Mathematical and numerical results are provided. Key Words: two-phase flows, eigenstructure, upwinding, perturbation methods.

1. Introduction In this paper, we examine the eigenstructure of a two-fluid model, derived from [TOU 96], when regularising terms are added to it. We think that such a study is of relevance to construct upwind schemes. Unfortunatly, it turns out to be a difficult task in two-fluid systems. Hence, several approaches are proposed to get practical approximations of the eigenelements : • A numerical algorithm. However, it may be expensive in CPU time for 3D finite volume calculations, although improvments are in progress ([ALO 98]). Also intensive use of symbolic calculation packages may help obtain closed form related expressions.

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• A perturbation method using a small parameter. We distinguish : — the density perturbation method to analyse the eigenstructure when the density ratio is small ( c = ^ < 1), see [COR& 98],[COR 98]. — Taylor expansion of the eigenvalues when the relative velocity is small compared to the speed of sound ( f = ^-

Two-fluid models are regularised by the addition of interface pressure correction terms [POK 97]. Indeed, in the basic model, surface tension effects are neglected, and it is assumed that pressure equilibrium exists between the two phases : p = pi — pg ^ Pi, with pi the interface pressure. Here, we give the standard homogeneous form of the physical set of conservation equations [TOU 96] :

where the subscript k refer to the gas phase ( k = g ) or to the liquid phase ( k — 1). otk is the volume fraction ( ag +&i = 1 ), pk the density, Vk the velocity, 6k the energy and h^ the enthalpy ( hk = &k + ^ )• Moreover, we note by Hk and Ek respectively the phasic total enthalpy and energy. For simplicity, we will consider a perfect gas. Several interface pressure correction models exist in the litterature. They must vanish when the velocities of the two phases become identical (vr = 0 ). We will apply the two perturbation methods to : • the pressure correction proposed in [LAH 92]

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• the pressure correction proposed in [TOU 96]

Actually, setting

we see that Cp = ag8 leads to correction (3) whereas 6 = 0 and Cp =• Cp(ag) leads to correction (2). We can compute the Jacobian matrix A of the two-fluid system :

As mentioned in the introduction, the interphase interactions yield a complex eigenstructure. Hence, perturbation methods may be a convenient way to get practical approximations of the eigenvalues and eigenvectors. 3. The two perturbation methods

3.1. The Taylor expansion of the eigenvalues In order to determine the eigenvalues of the system, we must find the roots of a polynomial of degree six P(A) given by :

A straightforward computation of this determinant leads to the following polynomial :

To get approximations of the roots of PA , one can use a perturbation method by introducing the ratio

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and the new variable sound in the two-phase mixture given by

with the 'characteristic' speed of

We look for a first order approximation of the eigenvalues and the eigenvectors (see [KUM 96]). Let the polynomial PA (A) be writen as

with We look for the roots of jP(z;£) in a neighbourhood of a root ZQ of the polynomial p o ( z ) . We will carefully distinguish the single roots and the double roots of Po(z}. Hence, introducing the notation 60 — 7 ^'^2 , we can find first related approximations of the two-fluid eigenvalues :

where k = I corresponds to the pressure correction (2) and k — ag correspond to the correction (3). Following the same method, the first order approximation of the eigenvectors can be derived (see [TOU 96]).

3.2. The density perturbation method In [COR& 98], it is shown that a scaling of the densities in the two-fluid system yields a splitting of the Jacobian matrix :

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t is the average density ratio e = p° /'p° and

7-A-i + AQ would represent the Jacobian matrix of a two-fluid system with no pressure gradient in the liquid phase. Hence, it is expected that its eigenelements are easy to compute. For instance, we get

The computations yield :

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In particular, when A-\ — 0, the perturbation theory for linear operators provides a convenient way to obtain higher approximations of the eigenelements of the original matrix ([HIN 91]). For instance the correction (2) yields A-i = 0 and in these circumstances, we can expand the eigenelements into a first order approximation in e ( [COR& 98]).

4. Numerical results The presence of non-conservative products yield significant difficulties. Here, we mainly follow [KUM 96]. Hence we construct the Roe type numerical scheme as follows :

with the numerical flux given by

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The positive and negative part of the Roe-averaged matrix are obtained thanks to first order approximations of the eigenstructure above. Some two-phase shock tube tests are computed using the two perturbation methods described in 3 and the two pressure corrections mentioned in 2. Moreover, we take 7 = 1.093 and pi = 720. The initial conditions for these shock tube problems are : TEST 1 Left state Right state

0.25 0.10

TEST 2 Left state Right state

ag 0.8 0.8

Oig

vg (m/s) 0 0 vg (m/s) 150 100

vi (m/s) 1 1 vi (m/s) 5 1

p (MPa) hg (kJ/kg) 3092.7 20 3099.9 10 p (MPa) 0.15 0.1

hi (kJ/kg) 1338.2 1343.4

hg (kJ/kg)

hi (kJ/kg)

12 12

3 3

The CFL number is 0.9 and the computations have been done with 400 cells. Results are provided after 200 iterations. The theorical results obtained in the previous section and the numerical results obtained in this one will be fully discussed in the presentation.

[ALO 98] F. ALOUGES, Une methode iterative pour decentrer sans diagonaliser, private communication [COR& 98] J. CORTES, A. DEBUSSCHE AND I. TOUMI, A Density Perturbation Method to study the Eigenstructure of Two-Phase Flow Equation Systems, Journal of Comp. Physics, Vol.147, No.2, 1998, pp. 463-484 [COR 98] J. CORTES, An Asymptotic Two-Fluid Model for Roe-Scheme Computation, Computational Fluid Dynamics ( ECCOMAS'98 proceedings ), John Wiley & Sons, Ltd., Vol.11, 1998, pp. 416-422 [EYM 99] R .EYMARD, T. GALLOUET, R. HEREIN, Finite Volumes Methods, to appear in the Handbook for Num. Anal., eds C. Lions, North Holland [GOD 91] E. GODLEWSKI, P . A. RAVIART, Hyperbolic systems of conservation laws, Mathematiques &; Applications, Ellipses, 1991 [HIN 91] E. J. HlNCH, Perturbation Methods, Cambridge Texts in App. Math., 1991 [KUM 96] I. TOUMI, A. KUMBARO, An approximate Linearized Riemann Solver for a Two-Fluid Model, Journal of Comp. Physics, Vol.124, 1996, pp. 286-300 [LAH 92] R. T. LAHEY, JR., The prediction of phase distribution and separation phenomena using two-fluid models, Boiling Heat Transfert, Elsevier Science Publishers, 1992, pp. 85-121

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[POK 97] H. POKHARNA, M. MORI AND V. H. RANSOM, Regularization of TwoPhase Flow Models : A comparison of Numerical and Differential Approaches, Journal of Comp. Physics, Vol. 134, 1997, pp.282-295 [ROE 81] P. L. ROE, Approximate Riemann solvers, parameter vectors and difference schemes, Journal of Comp. Physics, Vol.43, 1981, pp. 357-372 [TOU 96] I. TOUMI, An Upwind Numerical Method for Two-Fluid Two-Phase Flow Models, Nuclear Science and Engineering, Vol.123, 1996, pp. 147-168 [TIS 97]

I. TISELJ, S. PETELIN, Modelling of Two-Phase Flow with Second-Order Accurate Scheme, Journal of Comp. Physics, Vol.136, 1997, pp. 503-521

Figure 1: Test 1. Eigenstructure obtained by an exact calculation and the two perturbation methods

Figure 2: Test 2. Eigenstructure obtained by the two perturbation methods

Finite volumes simulations in magnetohydrodynamics

M. Hughes, L. Leboucher, V. Bojarevics, K. Pericleous, M. Cross Centre for Numerical Modelling and Process Analysis University of Greenwich London SE18 6PF — UK

ABSTRACT Magnetohydrodynamics involves the interaction of magnetic fields with fluid flows, so that simulations in this area of physics and its applications couple the resolution of Navier-Stokes equations and Maxwell equations. While most simulations in fluid dynamics are performed with finite volumes, most of the electromagnetic computations are traditionally performed with finite elements. The straight forward temptation is to solve each problem separately using one of the available finite volumes and finite element software tools, and feed back the results of one code into the other. This method can work reasonably well when the flow does not influence the magnetic field, that is to say when the flow only is influenced by the magnetic field. Then the amount of data exchange between the two codes is affordable. However, if both fluid flow and magnetic field have an influence on one another, then the data exchange and iterative computation between the two systems becomes too large. In such a case it is preferable to solve both problem within the same code and, for simplicity, with the same method, i.e. finite volumes or finite elements. While electrical engineering finite element software often gives approximate conservation of electrical quantities without serious inconvenience, fluid dynamics codes are very sensitive to the conservation of different physical quantities, especially the conservation of mass. Therefore, a good option is to solve both fluid dynamics and electromagnetic equations with the same conservative method, i.e. the finite volume method. Examples of conservative discretisations and simulations of electromagnetic systems will be given at the conference. Key Words: MHD, magnetohydrodynamics,

finite-volumes.

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1. Introduction An electrically conducting fluid of conductivity cr and velocity u interacts with a magnetic field B when induced or imposed electric currents of density j create a Lorentz force j x B within the fluid. This force may either be used to improve some casting processes or have unwanted stabilizing or destabilizing effects on liquid metal flows. The Ohm law in a moving fluid submitted to an electric potential and a time-dependent vector potential A defines the current

The electric potential may be solved from the divergence of this law, assuming Coulomb's gauge V • A = 0 and the conservation of electric charges V • j = 0

The magnetic field can be solved from the curl of Ohm's law (1), from Maxwell's equation V x B = fj,QJ and from the identity V x V x B = —V 2 B which holds for any conservative field — V • B = 0.

This is the induction equation. Finally, the Navier-Stokes equation with the Lorentz force is

where p and v are the density and the kinematic viscosity of the fluid and where the pressure p is obtained from the divergence of this equation, taking into account the conservation of mass V u = 0.

2. Imposed magnetic fields Rewriting the induction equation in dimensionless form gives exactly the same equation with the exception that the magnetic diffusivity (HQ(T)~I is replaced with the inverse of the magnetic Reynolds number Rm = poauL. In case where this number is small, corresponding either to a low electrical conductivity (^o^)" 1 ; to a small velocity or to a system of small length scale L, the magnetic field is unaffected by the motion of the fluid. Then the induction B may be computed by any mean, e.g with an electrical engineering simulation software. The same software may be used to compute the electric potential

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Figure 1: magnetohydrodynamic velocity field of the moten aluminium in an electrolysis cell. Stirring of the liquid metal by the electromagnetic forces.

and the resulting currents and Lorentz force. This force may then be added as a source term in any other computational fluid dynamics code. This is done easily with the finite-volume simulation software PHYSICA [PHY 99] where the electric currents and magnetic fields were taken from the finite-element code CADEMA [CHI 98] in order to solve the magnetohydrodynamic flow of a molten aluminium pad lying on an electrolysis cell of an aluminium production plant. Alternatively, once the magnetic and velocity fields are known, the electric potential may be solved from eq. (2) within the fluid dynamics code using the same finite volume method as for the pressure equation (5). This method is used to model another phenomenon of the same aluminium production process: the instability of the position of the interface between the aluminium pad and the electrolyte layer lying upon it. This instability is very similar to the waves at the surface of oceans and rivers apart from the fact that it is submitted to an electromagnetic perturbation coupled to the variations of the interface position. Here only the perturbation to the electric potential is computed within the fluid dynamics code [CHI 98]. The finite volume mesh is shown on figure 2. Note that the electric potential $ is shifted from the other scalar variable 77 denoting the height of the interface, in order to obtain the value of its derivatives d<&/dx and d^/dy at the positions where the electric currents and the resulting Lorentz force need to be known, i.e. at the positions of the momentum variables U and V. An example where all electromagnetic quantities are part of the fluid dynamics finite-volume code is the flow of a liquid metal in a rectangular

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Figure 2: finite volume mesh for magnetohydrodynamic system formulated in terms of the fluid variables U, V, TJ and the electric potential $. Note the position of

pipe [HUG 94] [LEB 99] submitted to a constant magnetic field. Figure 3 shows the electric currents induced by the magnetic field in the fluid moving perpendicularly to the plane of the figure. 3. Induced magnetic fields

When the magnetic Reynolds number is not small compared to 1, then the magnetic field is affected by the velocity field according to eq. (3). This

Figure 3: electric currents in the cross section of an electrically insulating duct with increasing magnetic field strength from left to right.

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Figure 4: iterative solution of Navier-Stokes and Maxwell's equations.

equation may be rewritten as a convection-diffusion equation

It appears then that the induction equation may be discretised and solved with the same finite difference scheme as the Navier-Stokes equation (4). The induction equation could however be solved by any electrical engineering finite element code. But exchanging the velocity and magnetic fields between the fluid dynamics and the electrical codes until both fields have converged may require too much time, especially if the data need to be read and written on the disk between each call to one of the two codes. Another reason for solving the induction equation with finite volumes is the conservative properties of this method. Some finite elements codes don't satisfy V • B = 0. This may not be a problem for steady electrical engineering simulations, but it becomes a serious drawback when time dependent simulations or iterative computation of fluid velocity and magnetic fields are performed. Indeed the error in V • B can amplify with time or with the iteration procedure, leading to completely wrong results or even to numerical instabilities. An example of magnetohydrodynamic channel flow where the magnetic field is convected by a large value of the fluid velocity is shown on figure 4. The Navier-Stokes and induction equations are solved iteratively with the same finite volume method. 4. Conclusion The nature of magnetohydrodynamics where all vector fields u, B and j are divergence free together with the dynamical properties of fluid flows makes the finite volume method suit particularly well to coupled electromagnetic and fluid mechanical problems. The same finite difference scheme can be applied to

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both of them, making magnetohydrodynamic codes shorter, simpler and more accurate than classical finite elements. 5. Bibliography [PHY 99]

http://physica.gre.ac.uk

[CHI 98]

CHIAMPI, M., REPETTO, M., CHECHURIN, V., KALIMOV, A., LEBOUCHER, L., 8th International IGTE Symposium on Numerical Field Calculation in Electrical Engineering, Institut fiir Grundlagen und Theorie der Elektrotechnik, Graz, Austria, September 1998.

[HUG 94]

HUGHES, M., Computational Magnetohydrodynamics, thesis, University of Greenwich, 1994.

[LEB 99]

LEBOUCHER, L., Monotone Scheme and Boundary Conditions for Finite Volume Simulation of Magnetohydrodynamic Internal Flows at High Hartmann Number, J. Comput. Phys., Vol. 150, pp.181-198, 1999.

Finite Volume Method for Large Deformation with Linear Hypoelastic Materials

K. Maneeratana and A. Ivankovic Department of Mechanical Engineering Imperial College of Science, Technology and Medicine, London, UK

ABSTRACT This paper describes a development of the finite volume method for modelling of structural problems involving geometrical non-linearities. The rate of Green strain tensor and second Piola-Kirchhoff stress tensor are used as the work conjugate pair in total and updated Lagrangian descriptions whilst the deformation rate tensor and Cauchy stress tensor with Jaumann objective stress rate are employed in the deformed Lagrangian (Eulerian) description. The law of conservation of linear momentum is discretised and resulting systems of algebraic equations are solved, using an iterative segregated procedure. The method is applied to a number of test cases and the results are shown to be in good agreement with analytical results. The accuracy, simplicity and adaptability of the method for non-linear structural problems are clearly demonstrated. Key Words: Finite Volume method, large deformation, hypoelastic materials.

\. Introduction This work concentrates on the development of the cell-centred FV method to geometrically non-linear problems involving linear hypoelastic materials. Since the method is inherently iterative, the computational effort for solving non-linearity is not expected to be vastly difficult. The law of conservation of linear momentum in the integral form is employed as the governing equation with linear hypoelastic materials as the constitutive relationships or equations of state. In the total strain or deformation theory (DT) scheme, the Green strain and second Piola-Kirchhoff stress are used. As hypoelastic constitutive equations have to express relationships between stress and strain rates,

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solids described by total strain formulation are, therefore, not hypoelastic materials. However, the St. Venant-Kirchhoff hyperelastic model, which has similar constitutive relationships to hypoelastic materials, are employed in this study [BON 97]. In the incremental approach, however, there are several possible schemes. The total Lagrangian (TL) and updated Lagrangian (UL) schemes are based on the initial and updated initial configurations, respectively. In both schemes, the rates of Green strain and second Piola-Kirchhoff stress are used as the work conjugate pairs. In the deformed Lagrangian (DL) scheme, the rate of deformation and Cauchy stress with Jaumann objective stress rate are employed. This scheme is also commonly called Eulerian [MCM 75] even though the formulations are not truly Eulerian as conventionally defined. In the incremental schemes, two methods of obtaining incremental governing and constitutive equations are considered. The derivation approach, denoted by 'deri' in the rest of the paper, simply differentiates the relationships with respect to time, while the total difference approach, denoted 'diff, considers the difference of total forces and deformations in the control volume. The governing equations are discretised by a cell-centred finite volume technique in line with earlier studies [DEM 94], [WEL98]. The presented method is, then, employed to solve simple problems and numerical results are compared with analytical solutions.

2. Governing and Constitutive Equations In this section, the governing and constitutive equations are briefly outlined. Details can be found in most continuum mechanics textbooks, such as [CRI 91] and [BAT 96]. Position of a particle within a body at a reference time t0 is defined by position vector x0. A motion, caused by applied forces, brings the particle to a new location x at a subsequent time t. In a TL formulation, all static and kinematic variables are referred to the initial configuration at time t0. Even though the formulation for DT scheme can be categorised in this approach, the abbreviation TL will be specifically referred to the incremental formulation hereafter for clarity. The UL formulation is based on the same procedure as TL scheme but variables are referred to the most recently calculated (updated) configuration. For the pure UL scheme, employed in this study, a new reference configuration xu is calculated for every increment of time or load. This updating is, in effect, resetting the displacement vectors to zero at the beginning of every increment and the TL formulations can be, therefore, simplified as will be shown in the next section. On the other hand, the deformed Lagrangian (DL) equations are based on the current deformed configurations x.

2.1. Governing

Equations

The basic law of conservation of momentum for the deformed body of volume v, bounded by deformed surface with normal vector a pointing outwards, is:

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where u is the displacement vector, CF is the Cauchy or true stress tensor, b is the body force, p is the current density and a dot above a quantity represents the time derivative of that quantity. As the Cauchy stress is based on the actual force, acting on deformed volumes, the second Piola-Kirchhoff stress tensor, S, is introduced as the mathematical mapping of the Cauchy stress to the undeformed volume as S — JF~l • o-(jP"1 )r where F = I + G is the deformation gradient tensor, G = du/dx0, I is the identity tensor and the Jacobian / = det(F). Therefore, equation [1] can be rearranged for DT approach for an undeformed body with initial surface vector a0, volume v0 and density p,, as:

where deformed surface vector a and volume v relate to a0 and v0 by da = J(F~>)7 -dao and dv = Jdvo. The equation [3] can be expressed in TL scheme using 'derivation' and 'difference' schemes as:

Equations [3] and [4] can be simplified for UL formulations to:

where the second Piola-Kirchhoff stress Su, rate of deformation gradient Fu, density pu, surface vector au and volume vu are referred to the last updated configuration. The reference position vector xu is updated every increment by setting it equal to jc of the previous increment, giving Fu = I since the tensor Fu is now based on the updated configuration. The rate form of equation [1] can be written for the DL formulation, using simple time 'derivation' and 'differencing' schemes as:

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Finite volumes for complex applications

where da - (tr(L)7 - (L) r ) • da and velocity gradient L = du/dx. If the force difference approach is used, the rate form of DL formation can be expressed by:

2.2. Constitutive

Equations

In order to obtain a well-posed system, the relationship between stress and strain measures for a given material is required. For the DT scheme, the Green strain E can be expressed using the deformation gradient F as:

The linear relationship between second Piola-Kirchhoff stress S and Green strain E can be expressed by:

where JJL and A are Lame constants. For the TL scheme, the rate form of equation [10] can be expressed as:

where the rate of Green strain obtained by simple 'differentiation' and 'different' schemes are:

For the UL scheme, the formulations of strain rate in equations [12] and [13] can be simplified to the Green strain rate in the updated configuration, Eu, as:

As UL and TL schemes are based on different configurations, constitutive relationship [11] must be corrected in order to obtain the equivalence between the TL and UL results [BAT 96]. The corrections is based on relationships between the TL and UL stress and strain rates as:

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This correction converts UL stress and strain tensors to the TL formulations but it also heavily complicates the expression. As such, there is no clear advantage of the UL formulations over the TL ones. However, if the strain is small, the equation [11] can be employed as the relationship between the rate of stress and strain measures for large deformation modelling. Therefore, this study employs this formulations for UL scheme with small strain restriction as:

Due to the change in reference configuration in UL scheme, the stress Su at the start of an increment is equal to (7 in the last increment. Therefore, the total value of Su is calculated from:

where n is the time step counter. In the DL formulation, the rate of (7 is frame different or not objective. As stress and strain measures in a constitutive relationship must be frame indifferent, an objective stress rate has to be employed. The rate of Cauchy stress a is related to the Jaumann objective stress rate G} by:

where W = |(L-L r ) is the rotation rate tensor and Z) = |(L + Lr) deformation rate.

2.3. Mathematical

is the

Model

By introducing constitutive relations into the governing equations, a general form of the transport equation is obtained in the form:

which can be regarded as an equation for the dependent variable 0, where 0 stands for displacement vector or displacement rate vector for total strain and incremental approaches, respectively, / ^ is the temporal coefficient, D $ is the diffusion coefficient, Q r

3. Discretisation Procedures The general governing equation [20] is discretised by employing a finite volume discretisation which has been successfully developed for CFD applications by

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Finite volumes for complex applications

[DEM 94] and [WEL 98]. The presented method takes the full advantage of control volumes of arbitrary shapes. It employs the fully implicit time discretisation and assumes piece-wise linear distribution of dependent variables and material properties in space. The temporal term is approximated by the mean value theorem. Surface and volume integrals are discretised using the mid-point rule. The non-orthogonality in the diffusion term is approximated by a simple orthogonal correction approach. The spatial discretisation is second order accurate while the temporal discretisation is only first order accurate. The resulting sets of coupled non-linear algebraic equations are segregated and the resulting algebraic systems of linear, de-coupled equations are iteratively solved by a pre-conditioning conjugate gradient method. 4. Test Case In order to demonstrate the method's accuracy and efficiency, few examples are modelled and numerical results are compared with analytical solutions in order to validify the code [MAN 99]. A problem presented here is a cube subjected to simple tension. The cube, with sides of length L, is discretised by 27 cubic Cvs. It is subjected to displacement V applied uniformly over one side while the opposite side is assumed to be symmetry plane. Other sides are free surfaces. The force F, required to deform the cube, is calculated from the resulting stresses on the loaded boundary. Material properties used in the calculation are E = 200 GPa and v = 0.3. Figure 1. shows the comparison between the numerical and analytical results. The analytical load - displacement solutions are based on the total Green strain and natural strain definitions [CRI 91]. For the solutions based on second PiolaKirchhoff stress and Green strain, the DT and TL-diff results coincide exactly while TL-deri results agree well with the analytical analysis. Results obtained by the UL schemes are shown to correlate with small strain solution. This is not surprising, as it is said in section 2.2., that the UL approaches in this study are restricted to large deformation problems under small strain conditions.

Figure 1. Displacement - force relationships for different

schemes.

Fields of application

Figure 2. Errors of F for different

465

schemes.

Similarly, DL results agree well with log strain solutions. Overall, the total difference schemes are more accurate than the simple derivation schemes (Figure 2.) because there are less problems of finite increment size used in the numerical procedures, which is assumed to approach zero in the rate equations. Comparisons of CPU times are shown in Figure 3. Times needed for these cases under small deformation in both total strain and incremental approachs are given as a guideline to the extra amount of time needed for large deformation problems.

Figure 3. Comparison of total CPU time required for different

schemes.

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Finite volumes for complex applications

5. Conclusions A numerical method for the analyses of a hypoelastic material is presented. The method is based on the FV discretisation of the law of conservation of linear momentum. Various work conjugate pairs of stress and strain measures are used in constitutive relationships. The developed technique is shown to be simple, easy to understand and manipulate since the underlying concept is simple and straightforward: the balance offerees in the solution domain. This procedure ensures both local and global conservativeness, resulting in meaningful solutions even for coarse numerical meshes. The non-linearity is handled with relatively small additional cost as the method is inherently iterative. This is achieved by using the segregated solution procedure which also offers very efficient memory management. It is clear that the DT approach is the most accurate but it is also limited to materials with single valued stress - strain curves (loading - unloading follow the same path) and is therefore not readily applicable to inelastic problems. The applicability of UL scheme is restricted to small strain cases while the DL approach with Jaumann stress rate is theoretically limited by since the Jaumann rate produces cyclic stresses under shear [KHA 95]. Thus, the TL scheme with 'difference' approach is considered to be the most flexible method; it is accurate and it can be further developed for inelasticity applications. Furthermore, this approach does not require expensive remeshing of the solution domain as the calculation progresses. So far, the method has been successfully employed for simple geometrically non-linear elastic problems. 6.

References

[BON 97] BONET J. et al., Nonlinear Continuum Mechanics for the Finite Element Analysis, Cambridge University Press, 1997. [MCM 75] McMEEKING R.M. et al., Finite-element formulations for problems of large elastic-plastic deformation, International Journal of Solids and Structures, 11, 1975, p. 601-616. [DEM 94] DEMIRDZIC I. et al., Finite volume method for stress analysis in complex domains, International Journal for Numerical Methods in Engineering, 37, N° 2, 1994, p. 3751-3766. [WEL98] WELLER H.G. et al., A tensorial approach to computational continuum mechanics using object oriented techniques, Computers in Physics, 12, 1998, p. 620-631. [CRI 91] CRISFIELD M.A., Non-linear Finite Element Analysis of Solids and Structures volume 1, John Wiley & Sons, 1991. [BAT 96] BATHE K.-J., Finite Element Procedures, Prentice-Hall, 1996. [MAN 99] MANEERATANA K. et al., Finite volume method for geometrically non-linear stress analysis applications, The Seventh Annual Conference of the Association for Computational Mechanics in Engineering ACME'99 (1999), p. 117-120. [KHA 95] KHAN A.S. et al., Theory of Plasticity, John Wiley and Sons, 1995.

A finite volume formulation for fluid-structure interaction. C.J. Greenshields, H.G. Weller and A. Ivankovic

Department of Mechanical Engineering Imperial College of Science, Technology & Medicine London SW7 2BX, UK

ABSTRACT The finite volume method has dominated computational fluid dynamics for many years and has recently emerged as a viable numerical method for stress analysis m solid structures. In this work we have adopted the technique for solving problems involving interaction between fluids and structures. The use of a single code enables simple transfer of information at the fluidstructure interface and an implicit solution procedure for the whole system within each time step. The method is used to study wave propagation in liquid filled, flexible pipes where, since the pressure wave is influenced by the flexibility of the pipe wall, the solution algorithm must be strongly coupled to obtain convergence. Key Words: finite volume; fluid-structure interaction; coupling.

1. Introduction Recent decades have witnessed the development of computational methods to solve problems involving interaction between solid bodies and fluids. Fluidstructure interaction (FSI) covers a wide range of engineering disciplines but a large proportion of all studies fall into the categories of flow-induced vibration, noise and hydrodynamics. Given such a limited scope of application, there is surprisingly little conformity in the numerical methods and coupling procedures adopted throughout research and industry [HAM 95]. The required level of fluid-structure solution coupling depends on the nature of the problem we want to solve. If the deformation of the structure is significant, we would preferably like to solve both components implicitly within each time step. At present, implicit coupling appears hindered by limited availability of a general purpose method capable of treating a variety of problems and by the complexity of

468

Finite volumes for complex applications

the methods currently practised. However, the finite volume (FV) method has recently emerged from the realms of computational fluid dynamics (CFD) as a viable alternative for structural analysis [DEM 93;DEM 94]. This presents a new possibility for FSI in which both fluids and structures are solved using the same approach [DEM 95]. The aim of this paper is to illustrate the simplicity and effectiveness of the FV method for FSI modelling through the analysis of wave propagation in liquid filled, flexible pipes. 2. Mathematical Formulation

This work is concerned with a simple mathematical description of the fluid and solid in which energy contributions are ignored. The continuum is governed by the following: • mass balance, or continuity equation

• momentum balance (neglecting body forces)

where p is the density, V is the velocity and cr is the stress tensor. In the case of a linear elastic (Hookean) solid, the continuity equation need not be considered and since deformations are sufficiently small, the convection term in the momentum equation can be ignored and V becomes d\J/dt, where U is displacement. The momentum equation then becomes:

where // and A are Lame's coefficients and I is the identity tensor. For a Newtonian fluid, the continuity equation must be rigidly obeyed and the momentum equation becomes:

where 77 is the fluid dynamic viscosity and p is the pressure. In this work, the system of fluid equations is closed by relating pressure and density for

Fields of application

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compressible flow in liquids by a linearised barotropic relationship

where the subscript '0' represents reference values and K is the bulk modulus. The specification of a problem is completed with the definition of a solution domain and initial and boundary conditions. At the initial instant of time, values of all dependent variables must be specified throughout the solution domain and boundary conditions must be specified at all times either by the value of the dependent variable or the gradient of the dependent variable. 3. Finite volume discretisation

The FV method is based on numerical integration of the system of equations over spatial and temporal domains and can be found in most CFD textbooks [PER 96]. Notable details of the method adopted here are: • face variables in the convection term are calculated using the Gamma differencing scheme [JAS 98] which locally blends second-order accurate central differencing with unconditionally bounded upwind differencing to maintain boundedness; • the PISO algorithm [ISS 96] is adopted to ensure that the velocity field in the momentum equation satisfies the continuity equation. The described discretisation procedure reduces both fluid and solid motion to a set of linear vector equations which are solved in a segregated manner, where each component of the dependent variable is solved separately, treating inter-component coupling terms explicitly. The structure displacements are solved using the Incomplete Cholesky Conjugate Gradient method and the fluid velocity and pressure are solved using the Biconjugate Gradient method [HAG 81]. 4. Fluid-structure coupling

The discretisation method described above has been implemented into the Field Operation and Manipulation C++ library [WEL 98]. The solid and liquid models are combined within a single code to model the transient behaviour of a flexible pipe and a contained compressible liquid. The fluid and structural parts of the solution domain form separate meshes but the interface boundary share

470

Finite volumes for complex applications

Figure 1: Implicit solution scheme for fluid-structure interaction the same location in space. The respective systems of equations are solved for both meshes and boundary conditions are passed as shown in Figure 1. 5. Fluid transients in water filled pipes

The test case concerns wave propagation in a water filled unplasticised PVC (PVCu) pipe of external diameter, D = 50 mm and wall thickness, t = 3.8 mm. The dynamic modulus, E — 4.0 GPa and Poissons ratio, v = 0.32 were measured at 20 °C for the PVCu material used in experimental work and its density was 1420 kg/m 3 . The properties of water used in the calculations were K = 2.2 GPa and = 998.2 kg/m 3 . The first simulation looks at the wave speed through water in a straight 200 mm length of pipe. The problem is axi-symmetric and the solution domain is 500 cells in the axial direction by 20 across the fluid and 8 across the wall section; the fluid is given an initial absolute pressure of 2 bar and zero velocity, and is contained at both ends of the pipe. The simulation begins by applying a fluid outlet condition of fixed 1 bar pressure at one end of the pipe. The first 50 //s of wave propagation is presented in figure 2 as pressure along the

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Figure 2: Wave propagation along a flexible pipe pipe axis. At 10 //s the fluid does not see the pipe wall deformation and, as expected, the initial wave travels at the speed of sound in unconfined fluids, c0 = \f~K~J~P — 1485 m/s. By 20 ^zs, the fluid pressure behind the unconfined wave front begins to rise in response to the pipe wall contraction. The initial unconfined wave is quickly suppressed by the surrounding regions of high pressure, as observed in experimentally [THO 69], and a true water hammer wave begins to propagate at an axial speed is 505 m/s. The classical analytical solution for fluid wave speed in an unconstrained, thick-walled pipe [WYL 93] gives a comparable speed of 546 m/s. The analysis only accounts for the change in pipe area which leads to an overprediction in wave speed of 7.5% compared to our numerical solution. 6. Solution stability and convergence

The efficacy of any iterative method is limited by the rate of convergence and worse still, by numerical instability. The coupling procedure adopted here

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Finite volumes for complex applications

Figure 3: Stability of the fluid-structure coupling procedure uses an outer iterative loop over the fluid and structure solver routines whose stability is examined by following the principal steps within a single iterative loop. Initially, the fluid equations are solved on the fluid mesh which has been deformed by the motion of the structure boundary within the previous outer iteration, provoking a change in pressure at the boundary, A.p. The structure equations are then solved and the cells adjacent to the boundary respond to Ap by a displacement Ub- The fluid mesh is then deformed once again and the process is repeated. A condition for stability is that |Ap| must tend to zero with successive iterations, which is illustrated schematically in figure 3. The pressure change at the boundary between successive iterations is depicted as a linear function of the change in displacement for both the fluid and structure. It is clear that if the structure cells are more rigid than the fluid cells at the boundary, the solution stabilises with successive iterations; if the fluid cells are more rigid, the procedure is unstable. In practice, the method is unstable for very flexible structures such as thin-walled plastic containers, hoses and arteries. In such cases, the instability can be eliminated by introducing a form of under-relaxation where, following the structure analysis, the fluid mesh is only deformed by a fraction a of the structural displacement. However, the approach is somewhat contrived since a must be carefully selected for the specific simulation under consideration to ensure reasonable convergence. Rapid convergence is only possible if there is little variation in cell size along the interface and therefore the method is only ideally suited to problems involving simple geometries and small deformations.

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7. Conclusions

Both fluid and structural analyses can be solved in a coupled, fully implicit manner using the FV method within a single code. The method has been validated using the analysis of wave propagation in liquid filled flexible pipes. In our 2 dimensional solution, there are only 5 unknowns and the coupling procedure adopted is easy to understand with the data transfer made simple by the use of one code and the same discretisation procedure for fluid and structure. The numerical prediction of wave speed and peak pressure give good agreement with analytical solutions and experiment and will improve with application of more accurate boundary conditions. Some under-relaxation is necessary to ensure stability of the coupling procedure in systems where the structure is comparatively more compliant than the fluid. The method can be applied to 3 dimensions with minimal additional effort and used to study problems such as water hammer in pipes, blood flow in arteries and impact of fluid filled containers.

Acknowledgements

The authors wish to thank Mr Tom van der Laan for the data from his pipe impact tests. CJG is currently funded by the EPSRC and HGW is funded by Computational Dynamics Ltd. References

[DEM 93]

DEMIRDZIC, I. & MARTINOVIC, D., Finite volume method for thermo-elastic-plastic stress analysis, Computer Methods in Applied Mechanics & Engineering (1993) 109 331-349.

[DEM 94]

DEMIRDZIC, I. fc MUZAFERIJA, S., Finite volume method for stress analysis in complex domains. International Journal for Numerical Methods in Engineering (1994) 37 3751-3766.

[DEM 95]

DEMIRDZIC, I. & MUZAFERIJA, S., Numerical method for coupled fluid flow, heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology. Computer Methods in Applied Mechanics & Engineering (1995) 125 235-255.

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[PER 96]

FERZIGER, J.H.

[HAG 81]

HAGEMAN, L.A. & YOUNG, D.M., Applied iterative methods. New York: Academic Press (1981).

[HAM 95]

HAMDAN, F.H. & DOWLING, P.J., Fluid-structure interaction: application to structures in an acoustic fluid medium, parti: an introduction to numerical treatment, Engineering Computations (1995) 12 749-758.

[ISS 86]

ISSA, R.I., Solution of the implicitly discretised fluid flow equations by operator-splitting, Journal of Computational Physics 62 40-65.

[JAS 98]

JASAK, H. et al, High resolution NVD differencing scheme for arbitrarily unstructured meshes, International Journal for Numerical Methods in Fluids in press.

[THO 69]

THORLEY, A.R.D., Pressure transients in hydraulic pipelines ASME Journal of Basic Engineering (1969) 91 453-461.

[WEL 98]

WELLER, H.G. et a/, A tensorial approach to continuum mechanics using object-oriented techniques, Computers in Physics (1998) 12 620-631.

[WYL 93]

WYLIE, E.B. & STREETER, V.L., Fluid transients in systems, Englewood Cliffs, New Jersey: Prentice Hall (1993).

BOUNDARY CONDITIONS FOR SUSPENDED SEDIMENT Vittorio Bovolin, Luca Taglialatela University of Salerno Department of Civil Engineering Via Ponte Don Melillo 184084 FISCIANO (SA) ITALY Ph. +39 089964087 Fax +39 089964045 Email bovo @ bridge, diima. unisa. it

ABSTRACT This paper deals with the problem of setting appropriate boundary condition for sediment concentration at the channel bed. Sediment transport is a common and very complex phenomenon, in many real situations suspended sediment represents the most important component of the total sediment load. In order to predict sediment load and morphological evolution numerical models are a very useful tool, application of these models requires the setting of boundary condition for the sediment transport equation. In this paper the boundary condition for sediment concentration is set according to a semiempirical relation that links the sediment flux to the local shear stress value. Calculations are carried out using a numerical model solving the averaged Navier-Stokes equations, the standard k-e model is used as turbulence model. Numerical results are compared with experimental data obtained in strong non uniform transport conditions such as: sediment pick-up by a flow initially free of sediment and sediment distribution in a steep sided trench. Key Words: sediment transport, turbulence, boundary condition

^^^^^

1. Introduction Sediment transport by turbulent flows is commonly encountered in practical applications such as rivers, navigation canals, estuaries and reservoirs. Sediments can be transported both as bed load and suspended load, the predominant mode of transport depends on the size, shape and density of the particles in respect to the velocity and turbulent field of the water body. The prediction of sediment transport with the aid of laboratory experiments may be very time-consuming and costly and, for many real life problems, impossible, hence there is a great need for powerful mathematical models to predict it. With the advance of computing methods in fluids,

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it is possible to obtain, at a reasonable computing time, numerical solutions for complex geometry. The boundary condition criteria for sediment concentration at the first grid node of the computational domain close to the bed are not as well established as are for the hydrodynamic counterparts. This paper tests a semi empirical condition proposed by [NAO 97], comparison of numerical result with experimental data shows good agreement.

2. Suspended sediment equation Sediments are subjected to two contradicting actions: velocity fluctuations lift up and keep them in suspension, gravity induce them to settle down. The upward flux of sediment due to the turbulent motion is vc, while the downward flux due to gravity is wsC, where v and c are the fluctuating components, respectively, of the velocity in the transverse direction and concentration, w, is the fall velocity and C is the local volumetric sediment concentration. Assuming an uniform sediment size and a single particle fall velocity ws. the differential transport equation for suspended sediment for steady 2-D turbulent open channel flow is given by:

where V and U are the mean velocity components in the orthogonal and in the flow direction. We consider only cases where the sediment concentration is so low that it does not interfere with the hydrodynamic. In these cases there is no coupling between flow and sediment equations and therefore equation [1] can be solved separately once the mean hydrodynamic field and the settling velocity are known, and when a model for the turbulent sediment diffusion vc has been adopted. In the present paper distribution of velocity, eddy diffusivity as well as friction velocity are calculate by numerical integration of the Reynolds equations with the standard k-e model as closure model. The computational procedure is based on the finite volume method using the SIMPLER algorithm for the pressure coupling [PAT 72], [LAU 74], [PAT 80], [ROD 80] and [VER 95]. Solution of the hydrodynamic model is used as a "frozen" in equation [1]. 3. Boundary condition In computational fluid dynamics it is customary to bridge the layer close to the wall by employing empirical formulae, called wall functions, these formulae provide near-wall boundary conditions for the mean-flow and turbulence transport equations [ROD 80], [VER 95]. These formulae therefore connect the wall conditions to the

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dependent variables at the near-wall grid node. The near-wall grid node is presumed to lie in the fully-turbulent part of the flow. The advantage of this approach is that it escapes the need to extend the computation down to the wall avoiding in this way the need to account for viscous effects in the turbulence model. The wall function approach also includes an option to permit wall roughness effects to be simulated via the specification of an equivalent "sand-grain" roughness height ks [JAY 69]. Once the hydrodynamic is known in order to solve equation [1] it is essential to set the appropriate boundary conditions at the bed. Generally speaking boundary condition can be of the fixed value type (Dirichlet), fixed flux type (Neumann) or mixed one. Fixing sediment concentration or flux at their equilibrium values may be an unrealistic option because, under non equilibrium condition, near-bed concentration and flux change drastically in the downstream direction before the equilibrium value is finally reached. It seems therefore essential to set mixed type boundary condition that may mimic sediment flux variation following changes in entrainment and deposition. Near the bed equation [1] reduces to [CEL 88] and [RAU 90]:

Integrating once yields:

left hand side of equation [3] represents the net flux J of sediment across an horizontal plane. The net flux J is the difference between the entrainment rate E and the deposition rate D. While the latter is equal to wsC the former is not immediately known and therefore it is the main problem in modelling suspended sediment transport in non-equilibrium condition. Starting from equation [3], applying the Reynolds analogy and mixing length concept, [NAO 97] developed a semi empirical boundary condition for uniform and non-uniform situations. Applying Reynolds analogy, [SHI 95] and [ROD 80] yelds:

where lm and /, are, respectively, the mixing lengths of momentum and particle concentration, being the velocity gradient multiplied by the mixing length proportional to the shear velocity, equation [4] can be rewritten as:

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where the proportionality coefficient crc, fixed by the authors, as suggested in literature, equal to 0.50 is known as the Schimdt number, substituting equation [5] in equation [3] a differential equation for C is obtained.

Integration of [6] in the range b

where z = (0cws )/(*"£/*) is the Rouse number and C/ the concentration at the first grid node. To overcome the need to evaluate Cb, it can be assess [NOA 97] that part of the turbulent eddies produced at the upper sublayer penetrates periodically the lower one and picks up sediments at a rate proportional to Q, max v'77, where v' is the root mean square of the vertical velocity fluctuation, replacing v' by £/*, the net flux is expressed as:

where 77 is the picking up efficiency assessed on theoretical basis:

and limited to d+=U*d/v>5. Evaluating Cb from equation [8] and substituting it in equation [7], the net flux J finally becomes:

Equation [10] is the general form for non uniform condition. The expression for uniform condition is obtained setting J=0 in [10], this is equivalent to fix the sediment concentration at the first grid node. Equation [10] can be used once the reference level b, and the concentration Cbjtnajc are specified. To evaluate the bed concentration we follow [RIJ 84] which gave the following expression:

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in which the mobility parameter Tis:

where a' is the standard deviation of effective bed-shear stress, TbiCr,i and Tb,crj are the instantaneous critical bed shear stress, respectively, along and counter local flow direction, J2 and J2 are integrals representing the pick up action of the bed shear stress. The non dimensional grain is:

in which D50 is the particle size, s is specific density (p/p), g the acceleration of gravity and v the kinematic viscosity coefficient (fj/p).

4. Results In order to test the proposed boundary condition we compare numerical results with experimental data obtained by [ASH 82] (as reported by [CEL 84]), by [RIJ 81] and [RIJ 85]. The first two data set ([ASH 82] and [RIJ 81]) refer to experiments in which initially clear water flow, passing over a sand layer, entrains sediment into suspension until the full transport capacity is reached. The second data set ([RIJ 85]) refers to measurements of sediment concentration profiles in steep sided trenches. Table 1 contains all the relevant information for the simulated cases. Case number Reference [RIJ 81] 1 RunT4

h

Um

d

ws (cm/s)))

B/h

Comments

0.230

2.80

0.02

Net entrainment from loose sand bed

k/h

m (mm))

47

0.03

(cm)

(cm/s)

25

2

[ASH 82] RunT5

4.3

37.35

0.15

0.165

1.85

0.15

Net entrainment sand on fixed bed

3

[RIJ 85] T2-T3

39

51

0.06

0.160

1.30

0.01

Net entrainment and deposition in trenches

Table 1. Relevant data for simulated cases

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For sake of simplicity following [RIJ 84] the reference level b has been set equal to O.Old where d is the flow depth. Inlet boundary (x=ff) are of fully developed channel flow. Comparison of calculated and measured sediment concentration profiles allows the assessment of the boundary condition capability to simulate localised effects, furthermore from a practical point of view it is also interesting to verify the capability of the model as a whole to reproduce the total suspended sediment discharge. Figure 1 depicts comparison of calculated sediment concentration profiles with experimental data [RIJ 81], [ASH 82], the general agreement is fairly good. Figure 2a, for the previous data, shows comparison of bottom sediment concentration, it gives an idea of the boundary condition capability to reproduce the pick up action by the flow, particularly noteworthy is the agreement between numerical results and experimental data in the first three sections where stronger is the flow non uniformity. In order to assess the model capability figure 2b reports comparison of calculated and measured total suspended load, also in this case the agreement is certainly encouraging on the model effectiveness to reproduce real life cases. The capability of the model to reproduce sediment concentration adjustment in case of non constant depth is another interesting test on the way to simulate morphological evolution. Figure 3 and 4 depict sediment concentration profiles and total suspended load for a steep sided trenches. This case is particularly demanding because the model has to simulate, in a short stretch, both net deposition, in the enlargement zone, where the flow decelerates and net entrainment in the zone where the flow is under a strong acceleration.

5. Conclusions This paper reports preliminary results of a study on boundary condition for sediment concentration. Numerical results shown fairly good agreement with experimental data. Further research is needed in order to assess the capability of the model to reproduce more complex situation as such as three dimensional case and morphological changes. 6. Acknowledgements The financial support for the research presented in this paper has been provided by a grant of the Italian Ministry for University and Scientific and Technological Research (M.U.R.S.T.) in the framework of the 1997 National Research Project (P.R.I.N.) "Swirling, turbulent and chaotic processes - Water works and Environmental applications".

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7. Bibliography [CEL 88] Celik I. and Rodi W., Modelling suspended sediment transport in non equilibrium situations, Journal of Hydraulic Engineering ASCE Vol. 114, No. 10 (1988) [JAY 69] Jayatilleke C. L. V., Tlie influence of Prandtl number and surface roughness on the resistance of the laminar sublayer to momentum and heath transfer. Prog. Heat Mass Transfer, Vol. 1, p. 193 (1969) [LAU 74] Launder B.E. and Spalding D.B., TJie numerical computation of turbulent flow, Comp. Meth. in Appl. Mech. & Eng., Vol.3, p269, (1974). [NAO 97] Naot D. and Nezu I., Wall functions for the calculation of turbulent 3D sediment transport in open channels XXVH IAHR Congress Vol. 2 San Francisco USA pp. 1268-1273(1997) [PAT 72] Patankar S. V. and Spalding D. B., A calculation procedure for heat, mass and momentum transfer in three dimensional parabolic flows, Int. J. Heat Mass Transfer Vol. 15 pp. 1787 (1972) [PAT 80] Patankar S. V., Numerical heat transfer and fluid flow, Hemisphere publising corporation, Taylor & Francis Group, New York (1980) [RAU 90] Raudkivi A.J., Loose Boundary Hydraulics, Pergamon Press, 3th edition (1990) [ROD 80] Rodi W., Turbulence models and their application in hydraulics, state-of-the-art paper. International Assoc. for Hydr. Res., Delft, The Netherlands (1980) [SHI 95] Shiono K., Falconer R. A., Berlamont J., Elzier M. and Karelse M., A note on statified flow in compound channel, Hydra 2000 XXVIth IAHR Congress, 3B8 pp. 134-139(1995) [RJJ 84] Van Rijn L.C., Sediment transport, part II: Suspended load transport Journal of Hydraulic Engineering ASCE, 110 (11), pp. 1613-1641 (1984) [RIJ 86] Van Rijn L.C., Mathematical modelling of suspended sediment in non-uniform flows Journal of Hydraulic Engineering ASCE, 112, pp. 433-455 (1986) [VER 95] Versteeg H. K. and Malalsekera W., An introduction to Computational Fluid Dynamics,the finite volume method, Longman Scientific & Technical (1995)

Figure L Sediment concentration profiles a) Ashida [ASH 82] b) van Rijn [RIJ 81]

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Figure 2. a) sediment concentration at the bottom b) suspended load [ASH 82]

Figure 3. Sediment concentration profiles in trenches, test2 andtestS [RIJ 85]

Figure 4. Suspended sediment load, respectively, in test2 and test3 [RIJ 85]

Second order corrections to the finite volume upwind scheme for the 2D Maxwell equations

B. Bidegaray

J.-M. Ghidaglia

MIP - CNRS UMR 5640 Universite Paul Sabatier 118 route de Narbonne 31062 Toulouse Cedex France

CM LA - CNRS UMR 8635 ENS de Cachan 61 avenue du president Wilson 94235 Cachan Cedex France

ABSTRACT When computing solutions to Maxwell equations with finite volumes methods one often faces mesh dependent structures. We describe a way to add second order corrections to the finite volume upwind scheme for the 2D Maxwell equations that are designed to overcome this difficulty. This is done by using exact solutions to the wave equation for each component of the electromagnetic field. We illustrate the method with numerical results on simple test cases.

Key Words : 2D Maxwell equations, finite volumes, upwind scheme.

1. Introduction

When computing solutions to Maxwell equations with finite volumes methods one often faces mesh dependent structures. Indeed, fluxes are computed across edges that may have privileged directions in some parts of the computational domain. Our goal is to write a modification of the classical upwind finite volume method that handles this drawback.

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Here we more precisely address the 2D Maxwell equations in the TEZ polarization. Setting Q = (Qi,Q-2,Q3) — (Ex,Ey,Hz} they read

In an homogeneous domain (i.e. e and [L constant) each component of Q satisfies the wave equation

where c2£/z = 1. Both formulations are used to derive our scheme. 2. Leading order terms

The computational domain is decomposed into triangles. Integrating (2) on one of them, K, and on the time interval [0,t], one gets

The computation of IK

is performed through the usual up-

wind scheme (see e.g. [GHI 96]) for hyperbolic systems using formulation (1). More precisely equation (1) is written as Qt + div(AQ) = 0 and IK reads IK = — I JdK

A(n)Qdcr where n is the external unit normal vector on the boun-

dary of the triangle K, dK. The eigenvalues of A(n) are 0 and ±c. For the computation of IK this integral may be split in three contributions considering separately each edge of K. Then we decompose A(n)Q over the eigenvector basis of A(n) and associate Q = Q(K) = to eigenvalue c and Q = Q(-K") to eigenvalue —c where K is K's neighbor across the considered edge of K. This is the usual upwind scheme. This is a particular case of our scheme and we will use it to evaluate its performances.

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3. Second order corrections

The second part,

which happens to be

a second order correction, is computed using the exact solution of the wave equation (2) with initial condition 1 on triangle K and 0 elsewhere. This approach has already been used by Abgrall [ABG 94], Gilquin et al [GIL 91, GIL 94] and Cha'ira [CHA 95] for solving the Riemann problem for gas dynamics and writing Euler equations as a wave equation, but in our case we have no conformal invariance. insertion of UQ which is formally bound to disappear in the next differentiation ensures that we preserve a constant solution over space (and time) if the initial data is constant, which is the first step towards flux conservation. We notice that this function w is solution to the wave equation

with initial data WQ = 0, initial time derivative w\ = UQ — UQ(K] and right hand An exact solution to this wave equation is given by Kirchoff formulae

and H denotes the Heaviside function. Hence the computation of UK leads to multiple integrals on the edges of the triangles that read (C and C1 being equal or adjacent edges)

and

that may be expressed after tedious calculations by means of classical functions of the variables (lengths and angles in the triangle). We make an approximation here by computing only integrals over edges of K or its neighbors but not

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triangles that only have one vertex in common with K, otherwise the computation of UK would be exact. This part of the derivation of our scheme is specific to the 2D case since the fundamental solution G to the wave equation has different expressions in different dimensions. The ID case is easy to compute but is of no interest since there is no point in correcting direction dependent structures. After this modification the scheme is no longer a finite volume scheme : there is no exact balance of fluxes. The formula for UK shows some discontinuity at time and if we denote by a the angle between edges C and C when are distinct, a acute

a obtuse

W0

W These formulae are continuous for a — | and are a puzzle for computer algebra systems. To obtain explicit formulae and avoid numerical integration is a major advantage for numerical computations. This computation is performed with a condition on the size of triangles and the time step (that implies, in particular that no length appears in the final result). More general conditions may be taken into account but then the integration should be performed on more triangle edges. To take account of the discontinuity at time t = 0, we do not use the full correction but only a fraction of it given by a factor 0 6 [0,1]. This means that we make a balance between time t and time 0 where UK — 0. Since we deal with a second order correction this does not affect the consistence of the scheme, whatever the choice of 9 is. The CFL ratio and this parameter 0 are the two parameters to tune in order to obtain a "good" scheme. They are strongly linked in ID but may be chosen independently in 2D. Numerical results show that 9 = 0 is not the best choice with respect to norm conservations for example. But the best 9 also depends on the mesh and is therefore difficult to choose.

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4. Numerical results This work is still in progress and up to now only simple test cases have been performed. Different boundary conditions have been implemented (following [ENG 77, ENG 79, JOL 89], namely absorbing boundary conditions, perfect conducting surfaces and perfect reflecting boundaries. Different initial conditions and incident fields have also been tested. The main drawback of the second order correction is that it induces some extra dissipation. This is very easy to see on a ID equivalent of our correction. This dissipation is larger for a large 9. This leads us to choose 0 E [0, .3] for numerical simulations. The best conservation of norms is usually obtained in this interval. The following figure shows that for a test where the L2 and L00 norm is supposed to be conserved, the choice the upwind scheme without correction (9 = 0) is not the better scheme for norm conservation. On Fig. 1 only the L°° norm of Ex is represented. The different curves correspond to different meshes for the same test case. Curves are similar for other components of the field or the L2 norm. The fact that curves are decreasing is not generic but specific to this field in the computed case. Our correction seem to benefit finer meshes but criteria to adjust 9 to a particular mesh are not obvious to find. Since we only add a small perturbation to the original scheme we can not expect any real improvement of its main defaults, like phase shift. The initial purpose of the introduction of our UK was the correction of mesh dependent structures and we have tested the propagation of a wave front with different angle of incidence. For small angles (otherwise our boundary conditions have to be improved) we show a real improvement of the straightness of the front. This shown on Fig. 2 and 3.

5. Perspectives Perspectives of this work may be found in different directions. First we may go towards more realistic test cases and try to model real physical structures. A complete study of the case of an inhomogeneous media has to be performed. This would also contribute to more physical test. The 3D case is also of interest and formal calculations of the correction have to be derived in this context.

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6. Bibliography [ABG 94]

[CHA 95]

[CIO 93]

[ENG 77]

[ENG 79]

[GHI 96]

[GIL 91] [GIL 94]

[JOL 89]

ABGRALL R., «Approximation du probleme de Riemann vraiment multidimensionnel des equations d'Euler par une methode de type Roe (I) and (II)», C.R. Acad. Sci. Paris 319, 1994, p. 499-504 and 625-629. CHAIRA S., Sur la resolution numerique des equations d'Euler de la dynamique des gaz par des schemas multidimensionnels, PhD Thesis, Ecole Normale Superieure de Cachan, 1995. CIONI J.-P., FEZOUI L. AND STEVE H., «A parallel time-domain Maxwell solver using upwind schemes and triangular meshes», Rapport INRIA 1867, 1993. ENGQUIST B. AND MAJDA A., «Absorbing Boundary Conditions for the Numerical Simulation of Waves», Math. Comput, 31, 1977, p. 629-651. ENGQUIST B. AND MAJDA A., «Radiation Boundary Conditions for Acoustic and Elastic Wave Calculations», Commun. Pure Appl. Math., 32, 1979, p. 313-257. GHIDAGLIA J.-M., KUMBARO A. AND LE COQ G., «Une methode Volumes finis' a flux caracteristiques pour la resolution numerique des systemes hyperboliques de lois de conservation», C. R. Acad. Sci., seme I, 332, 1996, p. 981-988. GILQUIN H. AND LAURENS J., «Problemes de Riemann multidimensionnels pour les systemes hyperboliques lineaires», 1991. GILQUIN H., LAURENS J. AND ROSIER C., «The explicit solution of the bidimensional Riemann problem for the linearized gaz dynamics equations», ENS de Lyon, UMPA 133, 1994. JOLY P. AND MERCIER B., «Une nouvelle condition transparente d'ordre 2 pour les equations de Maxwell en dimension 3», Rapport INRIA, 1047, 1989.

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FIG. 1: Norm conservation The L°° norm is represented as a function of 9 and for different meshes, the exact L2 norm being the horizontal line. The 'cross'-curve correspond to the finer irregular mesh (others are 'star' and 'circle'-curves) and the 'diamond'curve is a regular mesh that is finer than the 'square'-curve.

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FIG. 3: Front straightness : 0 — 0.1

A MHD—Simulation in Solar Physics A. Dedner^, C. Rohde^, M. Wesenberg^ t Institut fur Angewandte Mathematik, Universitdt Freiburg, Germany * Centre de Mathematiques Appliquees, Ecole Polytechnique, Palaiseau, France

ABSTRACT A Riemann-solver-based FV-scheme for the numerical solution of the equations of magnetohydrodynamics(MHD) in two and three space dimensions is proposed. Using recently developed techniques for the efficient combination of high-order resolution, adaptive mesh refinement and parallel computing the method is designed for complex flow problems in solar physics. Here we consider the rise of patterns of concentrated magnetic fields in the sun's convection zone. While rising to the solar surface these so-called flux tubes are fragmented by instabilities of Kelvin-Helmholtz- and Ray leigh-Taylor-type. Key Words: MHD, Riemann solver, high-order FV-schemes, adaptive mesh refinement, magnetohydrodynamical instabilities.

1. Introduction In the last decade the numerical solution of hyperbolic and hyperbolicparabolic conservation laws made considerable progress. In particular this applies to the case of several space dimensions. Features like upwind techniques, shock capturing, and formally high order resolution are standard now, even on unstructured grids. Furthermore the concepts of adaptive mesh refinement using grid indicators respectively error estimators and grid alignment became widely used. On the level of model problems a lot of corresponding analytic results on convergence and rigorous error estimates are available. Concerning physically relevant problems it is noteworthy that these techniques —despite their general applicability— have been utilized merely for the system of gas dynamics. Up to now, less has been done for more complex situations. Motivated by problems of solar physics this paper adresses two issues. First we present a numerical scheme that extends the methods mentioned above to the equations of ideal magnetohydrodynamics (MHD) in Section 2. Specific difficulties caused by the structure of the MHD-system are discussed, for in-

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stance the divergence free condition for the magnetic field. The second issue of this paper is the presentation of numerical results for phys ically relevant two and threedimensional MHD-simulations. We start with th discussion of two basic problems: the Kelvin-Helmholtz instability and th Rayleigh-Taylor instability (Sections 3.1 and 3.2). Finally we present result of ongoing numerical studies for a problem in solar physics which is strong! influenced by these instabilities (Section 3.3). To be specific we study th question how far a so-called flux tube, that is a pattern of concentrated mag netic field, can rise to the solar surface. If a flux tube can rise far enough thi would support the conjecture that flux tubes cause the sun spots which can b observed from earth on the solar surface. 2. Numerical Solution of the MHD-System in 2D/3D 2.1. The MHD-System The MHD-system describes the interaction of the motion of a compressible electrically conducting fluid with a magnetic field. For a spatial domain Q C R3 it is given by[Ca65, Ch61]:

Here p ( x , t ) stands for density, u(x,t) for velocity with components ux,uy,uz, B(x, t) for the magnetic field with components Bx,By,Bz, and E for the total energy. For the pressure p ( x , t ] , the matrix P(x,t) — ( p i j ( x , t ) ) e R 3 x 3 is defined as

To close the system, we add the thermodynamical relations

where T(x,t} is the temperature and e = £(p,T) the internal energy. Note that, for g(x,t) e R, the effects of gravity are included in the system above. We recall that the MHD-system above can be written as a system of hyperbolic

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conservation laws with the vector of unknowns U := (p,pu,B,pE) 6 U C R8, fluxes F, G, H : u ->• R8 and source 5 : ft x R>0 x U -> R8:

Initial and boundary conditions will be specified when needed. 2.2. The Numerical Scheme

In this section we will present our scheme to solve the MHD-equations numerically. For sake of simplicity we will restrict the presentation to a polygonal domain 0 C M2 that is covered by a grid of triangles. A part of the grid is displayed in Figure 1 to introduce notations.

Figure 1. Triangular grid and Notations. For initial data Uo : 17 —>• U, the FV-scheme takes the form

Here Atn denotes the local time step constrained by a suitable CFL-condition and \Tj\ the area of Tj. As usual, the numerical flux gjl is thought to be an approximation of

where ni = ( n j l , . . . , n*-jl) E R8, i = 1, 2. The term results from integrating the conservation law with respect to the time interval and the triangle Tj. Using the rotational invariance of the MHD-system the integrand can be expressed solely in terms of the flux F:

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This reduction to a onedimensional setting allows us to use (approximate) Riemann solvers. Based on an extensive study of different solvers in the MHDcase we favor the one suggested by Dai and Woodward, since it offers a good compromise between computational effort and accuracy [We98]. For details on this scheme we refer to [DW95]. By the linear reconstruction technique of Durlowsky, Engquist and Osher and the use of a second order Runge-Kutta method the basic FV-scheme described above has been extended (formally) to second order. To allow local refinement and coarsening of the grid we have to introduce an error estimator or a grid indicator. Error estimators for nonlinear hyperbolic systems in multiple space dimensions seem to be out of reach for the moment. For the Euler equations density-based indicators are well-established. Note that, in addition to the density, the magnetic field has to be taken into account for MHD. Even in one space dimension there exist discontinuous solutions of the MHD-system (Alven waves, entropy wave) which do not undergo a jump in density and the magnetic field. Extending a grid-indicator for the Eulerequations we propose the following grid indicator for refinement respectively coarsening.

The triangle Tj has to be refined, if ref_ind(Tj) > ref_limit, and may be coarsened, if crs_ind(Tj) < crs_limit, where ref_limit, crsJimit > 0 are some threshold values. Another MHD-specific problem is caused by the condition that V-B has to vanish. While this property is conserved by exact solutions for initially divergence free magnetic fields, it is not automatically satisfied in numerical schemes. A disregard of this problem can lead to severe numerical problems [BB80]. As a remedy we add an artificial source term, which is proportional to V • B [As96, Po94]. In our experience this approach works well in many situations but seems to have merely a stabilizing effect and does not necessarily reduce the divergence error. In particular problems with stagnation points remain a challenge. Even the computational advantages, that we achieve by local adaptation and higher order schemes, do not suffice to resolve small scale structures using a one-processor machine. Therefore our 2D-Code has been implemented on a parallel computer with a shared memory architecture. This gave good speedup results for large grids. Unfortunately computers with shared memory architecture and a large number of processors are not easily available. In 3D we use a code which can be run on machines with distributed memory by using MPI to communicate. This code is based on a grid concept which was developed by

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Schupp [Sc99]. To gain still from local grid refinement a concept for dynamic load balancing is integrated. 3. Numerical Results

As outlined in the introduction of this paper our code is designed for the simulation of problems in solar physics. In particular we are interested in the rise of flux tubes in the sun's atmosphere. These strong magnetic fields are assumed to develop in the deeper convection zone. When they rise they are subject to instabilities which break them up and slow down their ascend. One type of instability is caused by the lower density in the interior of the tube compared to the density of the atmosphere through which it rises. This results in Rayleigh-Taylor instabilities which increase the boundary of the tube and decrease its rise to the solar surface. At the same time the sides of the tube are fragmented by Kelvin-Helmholtz instabilities due to the shear flow between the atmosphere and the rising tube.

Figure 2. The time evolution of a KH instability for t — 0.4,0.6,0.8,1.0 using 256218,475374, 571044, 727138 elements respectively. Starting with [Ch61] both types of instabilities have been studied thoroughly in the literature for their own sake. The results show that the amplification of small scale disturbances can be reduced or even suppressed by a magnetic field tangential to the separating interface of the fluids. The magnetic field can be

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used to increase the stability of the rising flux tube, which leads to a rise into higher regions of the atmosphere without being too strongly fragmented. This stabilization is obtained by twisting the magnetic field in the flux tube so that it is tangential at the boundary. For physical background we refer to [Sch79]. In our calculations we show the performance of the code for pure KelvinHelmholtz- and Rayleigh-Taylor instabilities. Then we present two results for the flux-tube problem: with and without stabilization by twisting. For a quantitative study consult [DRW99]. 3.2. Kelvin-Helmholtz Instability (KH) In the first problem we consider a KH instability in a rectangular domai n that is separated in two parts by a horizontal line. For the initial conditior in fJ, we take The velocity in x-direction is ux0 = ±5 in the upper/lower part of n. To start the instability we perturb uy periodically. As boundary conditions we choose periodic conditions for the vertical boundaries and outflow conditions for the horizontal boundaries. The results for the transport of the density with the calculated velocity field at different time levels are displayed in Figure 2. For visualization we use a special tool to treat vector fields[BR98]. Note that gravitation is not included.

Figure 3. RT instability at time t = 4.5 for Bx/ f4 = 0.00,0.01,0.05,0.1.

3.3. Rayleigh-Taylor Instability (RT) The second test case shows a perturbed horizontal layer of lower density in a stratified atmosphere. Under the influence of gravitation this layer rises. Because of a perturbation of the initial values it produces the typical " finger"structure of the RT-type instability. We show the announced stabilizing influence of a growing tangential magnetic field in Figure 3 (Displayed quantity:

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density). As boundary conditions we have periodic conditions for the vertical conditions. For the the horizontal boundaries and the construction of the atmosphere we refer to [DRW99]. 3.4. Rise of flux tubes to the solar surface We turn to calculations of a flux tube in 2D. In the top row of Figure 4 we show By for a flux tube with a magnetic concentration normal to the plane of calculation. This magnetic field cannot reduce the fragmentation through the instabilities so that the flux tube is strongly fragmented and hardly rises.

Figure 4. Flux tube in 2D. In the lower row of Figure 4 the calculation is repeated with a twisted magnetic field. This prevents the fragmentation and allows the flux tube to rise faster and higher than in the first calculation. Figure 5 shows a first calculation for a flux tube in three space dimensions.

Figure 5. Flux-tube in 3D for a fixed time t — 0.6. Top: i/z-plane for x = 0, bottom: xz-plane for y = 1, y = 3.3, y = 5.6.

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ACKNOWLEDGEMENTS The authors were partially supported by the DFGSchwerpunktprogramm "Analysis und Numerik von Erhaltungsgleichungen" and the EU-TMR research network for Hyperbolic Conservation Laws. Bibliography [As96]

ASLAN N., Two-Dimensional Solutions of MHD Equations with an Adapted Roe Method, J.Num.Meth.Fluids 23, 1996, p. 1211-1222.

[BR98]

Becker J., Rumpf M.,, Visualization of Timedependent Vector Fields by Texture Transport Methods, In Proceedings of the Eurographics Scientific Visualization Workshop '98, 1998.

[BB80]

BRACKBILL J.U., BARNES D.C. The effect of nonzero V • B on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys. 35, 1980, p. 426-430.

[Ca65]

CABANNES H., Magnetodynamique des fluides, Paris (1965).

[Ch61]

CHANDRASEKHAR S., Hydrodynamic and hydromagnetic stability, Oxford (1961).

[DW95]

DAI W., WOODWARD P.R., A Simple Riemann Solver and High-Order Godunov Schemes for Hyperbolic Systems of Conservation Laws, J.Comp.Phys. 121 (1), 1995, p. 51-65.

[DRW99]

DEDNER A., ROHDE C., WESENBERG M., A numerical study on magnetohydrodynamic instabilities in three space dimensions, in preparation.

[Po94]

POWELL K.G., An Approximate Riemann Solver for Magnetohydrodynamics (That Works in More than One Dimension), ICASE-Report 94-24.

[Sc99]

SCHUPP B., Entwicklung eines effizienten Verfahrens zur Simulation kompressibler Stromungen in 3D auf Parallelrechnern, PhD-thesis, Freiburg (1999).

[Sch79]

SCHUSSLER M., Magnetic Buoyancy Revisited - Analytical and Numerical Results for Rising Flux Tubes, Astronomy and Astrophysics 71, No. 1-2, 1979, p.79-91.

[We98]

WESENBERG, M., A Note on MHD-Riemann-Solvers, print (1999).

Pre-

A Zooming Technique for Wind Transport of Air Pollution* P.J.F. Berkvens, M.A. Botchev, W.M. Lioen, J.G. Verwer CWI, P.O. Box 94079 1090 GB Amsterdam, The Netherlands http://www.cwi.nl/ e-mail: [berkvens, botchev, waiter, janv]@cwi.nl ABSTRACT In air pollution dispersion models, typically systems of millions of equations that describe wind transport, chemistry and vertical mixing have to be integrated in time. To have more accurate results over specific fixed areas of interest—usually highly polluted areas with intensive emissions—a local grid refinement or zoom is often required. For the wind transport part of the models, i.e. for finite volume discretizations of the transport equation, we propose a zoom technique that is positive, mass-conservative and allows to use smaller time steps as enforced by the CFL restriction in the zoom regions only. KEY WORDS: finite volumes, advection schemes, local refinement, air pollution, high performance computations 1

Introduction

Mathematical problems often encountered in air pollution modelling are transportreaction problems of the form

where cs are the concentrations of m chemical species in the atmosphere. The species under consideration are not only pollutants, but all the main chemical substances present in the atmosphere. In real applications m lies between 25 and 100. The first term in (1) denotes the time rate of change of cs, the term V • (ucs) describes the transport of the species by a given wind field u. The term V(u, cs) on the right-hand side appears as a result of parameterization of transport processes not resolved on the *This work has been done within the program "Wiskunde Toegepast" ("Mathematics Applied") of NWO, the Netherlands Organization for Scientific Research, project no. 613-302-040. This work was supported by NCF, the National Computing Facilities Foundation, under Grant NRG 98.02.

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grid. It will be referred to as vertical mixing. Vertical mixing is often modelled by means of turbulent diffusion parameterization. The stiff reaction term R describes the chemical reactions among the species cs. Usually, (1) is discretized in space with a finite-volume technique. For typical grid resolutions and numbers of chemical species ra, (1) yields a system of millions of equations that has to be integrated in time. More background information on air pollution modelling can be found in the recent books [4, 11] and the survey [7] giving an overview of numerical techniques used in the field. 1.1 Model In air pollution models problem (1) is typically discretized in space with a finitevolume technique and then integrated in time with operator splitting. This means that the whole advance in time consists of separate advection, vertical mixing and chemistry advances. Chemistry and vertical mixing are integrated in time implicitly, to avoid a severe restriction on the time step At imposed by the chemistry-mixing stiffness. Advection is usually integrated in time explicitly. In a particular atmospheric dispersion model we are working with, a successor of [5], several modern advection schemes are used. The two basic schemes are the Slopes scheme [9] and the Split scheme, a third order flux-limited upwind scheme [6, 8]. Both schemes are onedimensional and applied with directional splitting. For the atmospheric application it is important that mass is conserved and concentrations remain positive (cs(t) ^ 0, t ^ 0). Both advection schemes mentioned have these properties on uniform grids, even for divergent flows, which occur in the atmosphere. Positivity of species concentrations is guaranteed for time-step size restrictions such that no negative air masses can occur during any one of the split advection steps, which is a natural constraint. 1.2 Why zooming? To capture local phenomena—as occurring over highly polluted areas—without increasing the cost too much, it is often desirable to have a nonuniform grid with local refinement (zooming) over the areas of interest. The local grid refinement or zoom methods have been subject of active research (see e.g. relatively early works [1, 3]). Most of the effort has been to develop powerful adaptive grid refinement. In our case, however, adaptivity is not needed and not even welcome. First of all, areas of interest are known in advance (that can be Europe, for example). Second, meteorological data as e.g. wind fields are often not available on a fine grid in the whole domain (see Figure 1). Third, the position of the zoom regions is often determined by the chemistry part of the model: for example, one has to refine in the regions with high emission activity. Thus, our goal is an efficient robust zoom algorithm for fixed zoom regions and we propose a simple strategy for this. 2 Zooming technique 2.1 Requirements for zooming Since mass conservation and positivity are important for the atmospheric application, we wish our zooming technique to preserve both these properties (something we have

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Figure 1: An example of zooming. Zoom regions are numbered. not managed to find in the literature). Due to the smaller size of the finite volumes, the CFL restriction is more severe in zoom regions and a smaller time step At needs to be taken. Positivity and mass conservation are more difficult to preserve if we want to have smaller step sizes in the zoom regions only, so that e.g. in a zoom region with refinement factor two, two time steps are to be done within each global advance in time. Our zooming technique gives a simple and reliable way to have smaller timestep sizes in the zoom regions only while preserving positivity and mass conservation across zoom interfaces. 2.2 How to zoom in ID We explain first how to perform zooming for the one-dimensional (ID) transport equation ct + (uc)x = 0 and then how to extend this approach to the three-dimensional (3D) case. We use advection schemes in the mass conservative flux formulation. Each cell i contains a mass fa being the integral of c over the zth cell volume. In 1D fluxes Fi+1/2 are calculated giving the amount of chemical species transported per time step between cells i and i + 1. They depend on the given air-mass fluxes determined from the velocity field and on the masses fa. The time advance has the form

where Fi ±1/2 are the fluxes at time level n. On the boundaries of the zoom region its advection scheme is adjusted to the scheme on the coarse level. As an example, consider the situation with refinement factor ref = 3 (see Figure 2). This adjustment consists of two elements. First, the three outermost cells of the zoom region are lumped together and considered to be one single cell (the interface cell). The tracer mass in the interface cell is assumed to be distributed uniformly (or in accordance with its slope for the slope scheme) over its fine grid subcells, and the fluxes between these subcells are never computed. Second, on the boundary of the zoom region we calculate the flux Fc that is coarse in both space and time. For each coarse time step, there are three corresponding time substeps that have to be done in the zoom region. The coarse boundary flux Fc is applied at

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Figure 2: 1D zooming: one time step tn —» tn+1 corresponds to three time substeps in the zoom region. The "coarse" flux Fc is applied on the first substep. once at the first fine time substep, for the second and for the third substep zero flux is taken at the boundary of the zoom region. It can be shown that this simple strategy guarantees mass conservation and positivity of the chemical species for many modern advection schemes including the Slopes and Split schemes (for more details see [2]). The time-step restriction for positivity is basically the same as in the uniform case, namely that the coarse air mass flux on the one wall plus the three fine air mass fluxes on the other wall do not take out more mass than there was in the cell. The strategy is also easily generalized to the multidimensional case, with a similar time-step restriction per direction. Another choice would be to equidivide Fc in three parts and apply them consecutively at the same times as the fine fluxes on the other side of the interface cell. In ID as well as in multidimensional cases this gives no longer a positive scheme, unless a more severe time-step restriction is enforced. The problem is that parts of the coarse flux, which was calculated with the old concentration field, are applied to a new concentration field. The latter may have changed in such a way that positivity is lost. 2.3 How to zoom in 3D We now give more details about our zooming algorithm in 3D. For simplicity reasons we restrict ourselves to block-shaped zoom regions in 3D. For our applications this is sufficiently general. A zoom region has interface cells all along its edges forming the boundaries with its parent region. The ID algorithm described above can then be applied in a split manner. The only complication as compared with the ID case is caused by the fact that the advection scheme is directionally split. We use a Strang (symmetric) splitting scheme consisting of six split steps as follows: X-Y-Z-Z-Y-X, where X denotes the application of the advection operator in the z-direction during half a time step, At/2, etc. For one triplet of advection steps—one in each direction—on the coarse level, e.g. X-Y-Z, in the zoom region we perform ref triplets of advection steps in the same and the opposite order, thus x-y-z-z-y-x in case ref = 2. Here x denotes the application of the advection operator in the re-direction during a quarter time step, At/4, etc. For example, in Figure 3 we show the sequence of the directional substeps in a zoom region with refinement factor 2. Clearly, Strang splitting order is preserved in the zoom region, which is important for accuracy reasons. The coarse fluxes computed by the parent at the boundary with the child region

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Figure 3: The sequence of advection steps in a coarse and a fine region with refinement factor 2. are used as boundary conditions for the child region and are applied at once at the first time substep per direction. After the zoom or child region has carried out its ref triplets, the concentration field in the child region is copied back to its parent after suitable coarsening. This then allows the parent region to advance another three directional substeps and to provide new boundary conditions for the child region. This 3D zooming algorithm can also be shown to be mass-conservative and positive for the advection schemes we are interested in. The complete description of our algorithm can be found in [2]. 3 Experiments With our tests we want to demonstrate that (i) our zooming technique gives an accuracy comparable with the accuracy one would get on the overall fine grid provided that important phenomena remain in the zoom regions; (ii) this accuracy is achieved with significantly less computational effort; (iii) even when important phenomena occur outside the zoom regions, our algorithm still gives a solution which is at least as accurate as one would get on the overall coarse grid. The situation (i) is quite common for atmospheric modelling since one of the typical needs for the local refinement is concentration of the emission sources in highly polluted areas. We note in passing that zooming also leads to a more economical memory use, compared to the overall fine grid. We present 2D numerical results obtained with our zoom implementation for the Slopes advection scheme. As a model problem for our experiments we take the solidbody rotation test commonly known as the Molenkamp-Crowley test [10]. In this test the advection equation on the (unit) sphere

is solved for the velocity field u = 27rcosa:sin?/, v = -27r sin x. Here x E [0; 2yr] and y E (—Tr/2; Tr/2) are the longitude and the latitude coordinates respectively. The chosen velocity field provides a solid-body nature of the air rotation over the sphere, so that after one full rotation (at time t = 1) the solution will coincide with the initial distribution c(x, y, 0). For the initial distribution in our tests we take a cone of height 1 centered at the point (x, y) = (37T/2,0). We stress that the cone provides a severe test for any advection scheme. The radius of the cone base is taken to be 21 grid cells with respect to the finest zoom region used. In Figure 4, we show the velocity field and the position of the zoom regions used in the tests. Overall, we have a nonuniform grid consisting of three regions: the coarsest,

504

Finite volumes for complex applications

Figure 4: The velocity field and the position of the zoom regions. with resolution Ax = Ay = 4.5° and two zoom regions refined with factors 2 and 6 (resolutions Ax = Ay = 2.25° and Az = Ay = 0.75° respectively). First of all we have compared the zoom grid solution with the solution obtained on the overall fine grid (with the same resolution as in the finest zoom region). As desired, it turned out that the difference between them is negligible as long as the moving cone remains in the zoom region. The difference in the solutions is explained by the fact that the order of the directional substeps in the zoom algorithm alternates (as in Figure 3) whereas it remains the same for the uniform grid algorithm. Performing one full rotation (0 ^ t ^ 1) on the nonuniform zoom grid means that the cone has to travel over the whole sphere, leaving and then entering again the zoom regions. Of course, with the local zoom regions fixed in space, i.e. not moving with the cone, globally one can not expect much better accuracy than on the overall coarse grid. This is confirmed in our tests. We have performed one full rotation on the overall coarse (Ax; = Ay = 4.5°), the zoom, and the overall fine (Ax = Ay = 0.75°) grids. In each case, we have measured accuracy by comparing the solution cn after one full rotation with the initial distribution c° (the exact solution would give no difference with c°). We have computed the following errors (representing minimum, maximum, scaled /2, mean and variance errors respectively):

where rj is the (i, j)-cell air mass divided by the total mass of the air (cells near the poles have less mass), and the sums and the minima/maxima are taken for all the grid cells within the finest zoom region. To be able to compare errors of different space resolutions, on the overall fine and on the zoom grids we computed the errors on the data coarsened up to the coarse grid resolution. We have summarized our accuracy observations in Table 1. In Figure 5 (left picture), we plot the zoom solution after the full rotation. As we

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Table 1: Errors. "Local error" means with the cone cone still being within the finest zoom region, "global error" means after one full rotation Errors fine zoom coarse

local: ^global same as fine

emin -5.9e-3 -1.8e-2 -1.9e-2

emax -3.1e-2 -0.12 -0.21

global: errO 2.5e-3 1.7e-2 3.3e-2

errl 1.4e-3 1.8e-3 2.6e-3

err2 -3.5e-5 -0.12 -0.15

Figure 5: The cone after the full rotation see, the cone shape is strongly deformed. This is not only because it has travelled through the grid that is 6 times coarser (we should emphasize that the resolution 4.5° is quite coarse for this test) but also because of the accuracy losses near the poles. For comparison reasons, at the right plot we present the solution for the same test but with half the grid size in the coarse region (in other words, cone is to travel through the grid that is only factor 3 coarser, with basic resolution Ax = Ay — 2.25°). As expected, deformation is significantly decreased. Finally, we comment briefly on the computational expenses. Rough estimates taking into account total number of grid cells and the CFL restriction on the time step size show that the computations on the overall fine grid would be 5.2 times more expensive than on the zoom grid. This is however a too optimistic speed-up estimate which does not take into account communication overhead for the zoom algorithm. With the current implementation (SGI workstation) we observed that our zoom algorithm is approximately 2.6 times faster than the uniform algorithm on the fine grid. However, code optimization has to be performed yet. 4 Conclusions We have presented a positive and mass-conservative local grid refinement (zoom) algorithm for advective transport. The algorithm can be applied to many modern advection schemes with directional splitting in space and explicit advance in time. With our approach, a smaller time step (due to the stricter CFL condition) is taken within the zoom regions only.

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Finite volumes for complex applications

The described zoom algorithm has been implemented as a code that allows to use an arbitrary number of zoom regions that can lie inside each other in an arbitrary way, provided they are of block shape and strictly embedded. To have high performance of the uniform grid code preserved as much as possible, the work of the zoom code is organized in a uniform region-wise manner, i.e. the zoom grid is split into a cascade of uniform grid regions. For more details on the implementation we again refer to [2]. Our zoom code will be incorporated in the air pollution model TM3, a recent successor of [5]. The TM3 model is operational at the Institute of Marine and Atmospheric Research (IMAU, Utrecht University), the Dutch Royal Meteorological Center (KNMI) and the Dutch National Institute of Public Health and the Environment (RIVM). References [1] M. J. Berger. Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations. PhD thesis, Department of Computer Science, Stanford University, Stanford, CA 94305, Aug. 1982. [2] P. J. F. Berkvens and M. A. Botchev, W. M. Lioen and J. G. Verwer. A zooming technique for wind transport of air pollution. Report MAS-R99xx, CWI, 1999. [3] J. H. Flahery, P. J. Paslow, M. S. Shephard, and J. D. Vasilakis, editors. Adaptive methods for Partial Differential Equations. SIAM, Philadelphia, PA, 1989. [4] T. E. Graedel and P. J. Crutzen. Atmosphere, Climate and Change. Scientific American Library. Freeman and Company, New York, 1995. [5] M. Heimann. The global atmospheric tracer model TM2. Technical Report 10, Deutches Klimarechenzentrum (DKRZ), Hamburg, 1995. [6] W. Hundsdorfer and B. Koren, M. van Loon and J. G. Verwer. A positive finitedifference advection scheme. Journal of Computational Physics, 117:35-46, 1995. [7] J. G. Verwer, W. Hundsdorfer and J. G. Blom. Numerical time integration for air pollution models. Report MAS-R9825, CWI, 1998. [8] A. C. Petersen, E. J. Spec, H. van Dop, and W. Hundsdorfer. An evaluation and intercomparison of four new advection schemes for use in global chemistry models. Journal of Geophysical Research, 103(D15):19,253-19,269, Aug. 1998. [9] G. L. Russell and J. A. Lerner. A new finite-differencing scheme for the tracer transport equation. J. Appl. Meteor., 20:1483-1498, 1981. [10] D. L. Williamson and P. J. Rasch. Two-dimensional semi-Lagrangian transport with shape-preserving interpolation. Monthly Weather Review, 117:102-129, 1989. [11] Z. Zlatev. Computer treatment of large air pollution models. Kluwer Academic Publishers, 1995.

Computational Solid Mechanics using a Vertex-based Finite Volume Method

G. A. Taylor, C. Bailey and M. Cross Centre for Numerical Modelling and Process Analysis University of Greenwich, Woolwich, London SE18 6PF, UK E-mail: [email protected]

ABSTRACT A number of research groups are now developing and using finite volume (FV) methods for computational solid mechanics (CSM). These methods are proving to be equivalent and in some cases superior to their finite element (FE) counterparts. In this paper we will describe a vertex-based FV method with arbitrarily structured meshes, for modelling the elasto-plastic deformation of solid materials undergoing small strains in complex geometries. Comparisons with traditional FE methods will be given. Key Words: Vertex-based, Finite Volume, Solid Mechanics, Elasto-plastic.

1. Introduction Over the last three decades the FE method has firmly established itself as the pioneering approach for problems in CSM, especially with regard to deformation problems involving non-linear material analysis [OH80, ZT89]. As a contemporary, the FV method has similarly established itself within the field of computational fluid dynamics (CFD) [PatSO, Hir88]. Both classes of methods integrate governing equations over pre-defined control volumes [PatSO, Zie95], which are associated with the elements making up the domain of interest. Additionally, both approaches can be classified as weighted residual methods where they differ with respect to the weighting functions that are adopted [OCZ94]. Over the last decade a number of researchers have applied FV methods to problems in CSM [Tay96]. It is possible to classify these methods into two approaches, cell-centred [DM92, HH95, Whe96, Whe99] and vertex-based [FBCL91, OCZ94, BC95, Tay96]. The first approach is based on traditional FV methods [Pat80] as applied to problems in CFD and suffers from the same difficulties when applied to complex geometries involving arbitrarily structured meshes [DM92, HH95]. The second approach is based on traditional

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Finite volumes for complex applications

FE methods [ZT89] and employs shape functions to describe the variation of a variable over an element, and is therefore well suited to complex geometries [FBCL91, OCZ94]. Both approaches apply strict conservation over a control volume and have demonstrated superiority over traditional FE methods with regard to accuracy [Whe96, Tay96], some researchers have attributed this to the local conservation of a variable as enforced by the control volumes employed [FBCL91, BC95] and others have attributed it to the enforced continuity of the derivatives of variables across cell boundaries [Whe96]. The objective of this paper is to describe the application of a vertex-based FV method to problems involving elasto-plastic deformation and provide a detailed comparison with a standard Galerkin FE method. 2. Equilibrium Equations and Boundary Conditions

In matrix form, the incremental equilibrium equations are where [L] is the differential operator, {Acr} is the Cauchy stress, {&} is the body force and n, is is the domain. The boundary conditions on the surface T = Tt U Fu of the domain ft can be defined as [ZT89, OCZ94]

where {tp} are the prescribed tractions on the boundary Ft, {up} are the prescribed displacements on the boundary Tu and [R] is the outward normal operator [OCZ94, Tay96]. 3. Constitutive Relationship

In matrix form, the stress is related to the elastic strain incrementally as follows; {Acr} = [_D]{Aee}, where [D] is the elasticity matrix. For the deformation of metals, the von-Mises yield criterion is employed and the elastic strain is given by {A6e} = {Ae} — {Aevp}, where {Ae} and {Ae^p} are the total and visco-plastic strain, respectively. The visco-plastic strain rate is given by the Perzyna model [Per66]

where aeq, ay, 7, N and s are the equivalent stress, yield stress, fluidity, strain rate sensitivity parameter and deviatoric stress, respectively. The < x > operator is defined as follows;

Fields of application

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The total infinitesimal strain is {Ae} = [L]{Aw}, where {Aw} is the incremental displacement.

4. Vertex-based Discretisation Employing the constitutive relationship of the previous section in equations (1) and ( 2 ) , and assuming the boundary conditions as described by equation (3) are directly satisfied by the vector {Aw}, the method of weighted residuals can be applied to the equations to obtain the following weak form of the equilibrium equation [ZT89];

where [W] is a diagonal matrix of arbitrary weighting functions. At this point the unknown displacement can be approximated as

where {Au}j is the unknown displacement at the vertex j, Nj is the shape function associated with the unknown displacement and [/] is the identity matrix. The displacement approximation can be introduced into equation (5) if the arbitrary weighting functions [W] are replaced by a finite set of prescribed functions [W] = £"=1[^],, for each vertex i [ZT89, OCZ94], /»

f

Equation (7) can be expressed as an incremental linear system of equations of the form [A'] {Aw} — {/} = {0}, where [K'] is the global stiffness matrix, {Aw} is the global displacement approximation and {/} is the global equivalent force vector and can be formed from the summation of the following contributions;

510

Finite volumes for complex applications

Figure 1: 2D control volumes, (a) overlapping FE and (b) non-overlapping FV. where £7; is the control volume associated with the vertex i and F; = F U i U Tti is the boundary of the control volume.

4.1. Standard Galerkm FE Method In the standard Galerkin FE method the weighting function associated with a vertex is equal to the shape function of the unknown associated with tha' vertex [ZT89, Hir88, OCZ94], [W], = [N]i. The shape functions describe the variation of an unknown over an element and there can be a number of elements associated with each vertex. Hence, it is apparent that control volume: described by weighting functions of this form will always overlap. This is illus trated in Figure l(a) for a simple two dimensional case of two adjacent node; i and j, where the control volumes fij and nj have contributions from all the elements associated with their respective vertices i and j. Hence, for the standard Galerkin FE method the contributions as describee by equations (8) and (9) are

where [B],- = [LN]i. It is important to note that if the boundary of the control volume, such as that described by F; in Figure l(a), coincides with the external boundary of the domain, the shape functions are not necessarily zero along that part of the boundary. Thus, if a flux is prescribed such as a traction this will not necessarily disappear and may contribute to the equivalent force vector as described in equation (11). Additionally, the symmetrical nature of the stiffness matrix as indicated by equation (10) should be noted. The Galerkin approach is

Fields of application

511

accepted as the optimum technique for treating physical situations described by self-adjoint differential equations, particularly those in solid mechanics, as the inherent symmetrical nature is preserved by the choice of weighting functions [ZT89, OCZ94]. 4.2. Vertex-based FV Method In the vertex-based FV method the weighting functions associated with a vertex are equal to unity within the control volume, [W]i = [/], and zero elsewhere. This definition is equivalent to that for the subdomain collocation method as defined in the standard texts [Hir88, ZT89]. Though it is important to note that weighting functions defined in this manner permit a variety of possibilities with regard to the control volume definition [OCZ94]. This is because the weighting functions are not restricted to to a direct association with the cell or element as in the Galerkin case. This is an important consideration and requires the recognition of the vertex-based FV method as a discretisation technique in its own right [Hir88]. For the vertex-based FV method the contributions as described by equations (8) and (9) are

It is important to note that the traction boundary conditions can be applied directly as another surface integral, but in the previous Galerkin approach an additional surface element is generally included on the domain boundary. A non-overlapping control volume definition suitable for a vertex-based FV method is illustrated in two and three dimensions in Figures l(b) and 2(a), respectively. The Figures illustrate the assembling of vertex-based control volumes from their required sub-control volumes [Tay96]. Additionally, the asymmetric nature of the contributions to the overall stiffness matrix as described by equation (12) does not ensure that symmetry will always be preserved. For this reason FV methods were initially argued as being inferior, but in the light of recent research where different control volume definitions have been proposed, the extent of this inferiority has come into question [OCZ94, Zie95, BC95]. 5. Results and Conclusions

In this section the vertex-based FV method is applied to a three dimensional validation problem and compared with the standard Galerkin FE method. The non-linear solution procedure adopted in for both these methods is based upon

512

Finite volumes for complex applications

Figure 2: (a) 3D assembly of FV sub-control volumes and (b) spherical vessel. that of Zienkiewicz and Cormeau [ZC74, Tay96]. Both methods utilised an explicit technique with regard to time stepping of the Perzyna equation (4). It is important to note that the FV solution procedure only differs from that of the FE in contributions to the global equivalent force vector and the global stiffness matrix. Hence, allowing an accurate comparison of the two methods [Tay96]. The methods are compared with regard to accuracy and computational cost. They are also analysed for a variety of meshes with different element assemblies. 5.1. Test case: Internally pressurised spherical vessel For this validation problem a thick walled spherical vessel, consisting of an elastic-perfectly plastic material, undergoes an instantaneously applied internal pressure load. The pressure load is 30 dNmrn" 2 , the Youngs modulus and Poisson ratio required to define the elasticity matrix are 21,000 d N m m - 2 and 0.3, respectively, and the yield stress is 24 dNmm- 2 . This problem is rate independent and the final solution is equivalent to that of an elasto-plastic analysis [ZC74]. A closed form radial solution is available [Hil50]. Numerically the problem can be modelled in three dimensional Cartesian coordinates, with the displacement components fixed to zero in the relative symmetry planes. The spherical vessel is then reduced to an octant as illustrated in Figure 2(b) 1 . Examples of meshes consisting of linear tetrahedral (LT), bilinear pentahedral (BLP) and trilinear hexahedral (TLH) elements are illustrated in Figures 2(b) 2 , 2(b) 3 and 2(b) 4 , respectively. Firstly, the problem was analysed with a series of meshes consisting of TLH elements. The hoop stress profiles, along the radii, as obtained from one of the numerical analyses are plotted and compared against the reference solution in Figure 3(a). The profiles illustrate the stress in the plastic and elastic regions, and the radial extent of the plastic region according to the analytical solution. The close agreement of the two methods is illustrated. However, it

Fields of application

513

Figure 3: (a) 950 TLH and (b) 4,800 LT elements. is important to note the closer agreement between the reference solution and the FV method when a coarse mesh is employed. These observations may be associated with the higher order, trilinear nature of the elements employed in the three dimensional analysis at this stage. With regard to the FV method, the implementation of pressure loads (tractions) will involve bilinear face elements for TLH elements. Hence, when considering the application of pressure loads for the two methods as described in equations (11) and (13), the contributions are different due to the individual weighting technique associated with each method. Furthermore, the weighting technique employed for the FV method may be more complementary, when applied generally, as all the terms are integrated conservatively at a local level. Conversely, for the FE method the weighting is not locally conservative which may introduce errors when pressure loads are employed. These conclusions are tentative and rely on the interpretation of the present observations, but they agree with the findings of other researchers [Whe96] and strongly suggest that further research of the FV method is worthwhile. Secondly, the problem was analysed with a series of meshes consisting of BLP elements and there was much closer agreement between the methods [Tay96]. This is attributable to the lower order, bilinear nature of the element concerned and the linear nature of the triangular faces over which the pressure loads were applied. As illustrated in Figure 2(b) 3 the BLP elements are orientated so the pressure load was prescribed over a triangular face. This was an outcome of the automatic mesh generator employed [Fern] and it is possible to further study the element when pressures are applied to the bilinear, quadrilateral faces, though it was not studied in that research. Thirdly, the problem was analysed with a series of meshes consisting of LT elements. The hoop stress profiles from one of the analyses are plotted in Figure 3(b). There is complete agreement between the methods with regard to LT elements as the global stiffness matrices and global force vectors constructed by the two methods are identical. This is a consequence of the linear nature of

514

Finite volumes for complex applications

Figure 4: (a) CPU times on a SPARC 4, 110MHz. both the element concerned and the triangular faces over which the pressure is applied. It is possible to demonstrate this equivalence analytically [Tay96] by extending to three dimensions, a two dimensional analysis which has been applied to elastic problems involving linear triangular elements [OCZ94]. Finally, the methods were compared with regard to computational cost. Considering LT elements, as the matrices are identical and symmetric a conjugate gradient method (CGM) is applicable in both cases. As illustrated in Figure 4(b), the FV method (FV-CGM) requires more CPU time than the FE method (FE-CGM) even when the same linear solver is employed. This is expected as the FV method visits six integration points, while the FE method visits a single Gauss point when adding contributions to the linear system of equations [Tay96]. Considering TLH elements, the geometrical nature of this validation problem prohibits an orthogonally assembled mesh. Hence, for the FV method a bi-conjugate gradient method (Bi-CGM) is required due to the asymmetric nature of the coefficient matrix obtained [Tay96j. Conversely, for the FE method a CGM is sufficient as the matrix obtained is symmetric. These requirements agree with the discussions in the previous section. As illustrated in Figure 4(a), the FV method (FV-BiCGM) requires approximately twice the CPU time as the FE method (FE-CGM). This is also expected due to the computational requirements of the two different linear solvers employed. Also for TLH elements, the FV method visits twelve integration points per element, while the FE method visits eight Gauss points per element. Hence, it can finally be concluded that any improvement in accuracy obtained by employing the vertex-based FV method must be offset against the extra computational cost required. Bibliography [BC95]

C. Bailey and M. Cross. A finite volume procedure to solve elastic solid

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mechanics problems in three dimensions on an unstructured mesh. Int. Journal for Num. Methods in Engg., 38:1757-1776, 1995. [DM92]

I. Demirdzic and D. Martinovic. Finite volume method for thermo-elastoplastic stress analysis. Computer Methods in Applied Mechanics and Engineering, 109:331-349, 1992.

[FBCL91] Y.D. Fryer, C. Bailey, M. Cross, and C.-H. Lai. A control volume procedure for solving the elastic stress-strain equations on an unstructured mesh. Appl. Math. Modelling, 15:639-645, 1991. [Fern]

Femview Ltd., Leicester, UK.

FEMGEN/FEMVIEW.

[HH95]

J.H. Hattel and P.N. Hansen. A control volume-based finite difference method for solving the equilibrium equations in terms of displacements. Appl. Math. Modelling, 19:210-243, 1995.

[HilSO]

R. Hill. The Mathematical Theory of Plasticity. Clarendon Press, Oxford, UK, 1950.

[Hir88]

C. Hirsch. Numerical Computation of Internal and External Flows: Fundamentals of Numerical Discretisation, volume 1. John Wiley and Sons, 1988.

[OCZ94]

E. Onate, M. Cervera, and O.C. Zienkiewicz. A finite volume format for structural mechanics. Int. Journal for Num. Methods in Engg., 37:181-201, 1994.

[OH80]

D.R.J. Owen and E. Hinton. Finite Elements in Plasticity: Theory and Practice. Pineridge Press Ltd., Swansea, UK, 1980.

[PatSO]

S.V. Patanker. Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington DC, 1980.

[Per66]

P. Perzyna. Fundamental problems in visco-plasticity. Advan. Appl. Mech., 9:243-377, 1966.

[Tay96]

G.A. Taylor. A Vertex Based Discretisation Scheme Applied to Material Non-linearity within a Multi-physics Finite Volume Framework. PhD thesis, The University of Greenwich, 1996.

[Whe96]

M.A. Wheel. A geometrically versatile finite volume formulation for plane elastostatic stress analysis. Journal of strain analysis, 31(2):111-116, 1996.

[Whe99]

M.A. Wheel. A mixed finite volume formulation for determining the small strain deformation of incompressible materials. Int. Journal for Num. Methods in Engg., 44:1843-1861, 1999.

[ZC74]

O.C. Zienkiewicz and I.C. Cormeau. Visco-plasticity—plasticity and creep in elastic solids—a unified numerical solution approach. Int. Journal for Num. Methods in Engg., 8:821-845, 1974.

[Zie95]

O.C. Zienkiewicz. Origins, milestones and directions of the finite element method - a personal view. Archives of computational methods in Engg., 2:1-48, 1995.

[ZT89]

O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method: Volume 1: Basic Formulation and Linear Problems. Magraw-Hill, Maidenhead, Berkshire, UK, 1989.

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Control volumes technique applied to gas dynamical problems in underground mines

Elena Vlasseva Associated Professor Department of Mine Ventilation and Labour Safety University ofMining&Geology "St. Ivan Rilski" Sofia 1100, Bulgaria

ABSTRACT. Paper presents an application of Control Volumes' Method in the field of underground mining. Modeling deals with noxious gas distribution (concentration) in time and space along mine roadway or network. Models take into account variable velocity, density and outside gas inflows in mass balance of air flow. Different source functions, which due to technological or accidental reasons, provoke occurrence of transient process, are presented in the paper. Numerical and computer models are verified with real data, obtained unfortunately from great disaster, which took place in one Bulgarian underground coal mine in 1997. Models serve as a tool used to reveal the circumstances lead to that disaster. Modeling results might be also applied to help introduction into practice of engineering solution, which can be analyzed due to their practical implementation. Key words: control volumes; gas distribution; underground mining

1.

Introduction

Air flows in underground mines are of great importance for human lives underground. Normally flows transfer not only fresh air but different impurities, liberated from rocks or as a result of mining operations. Keeping of these impurities into safety limits is one of the main task for underground staff. Once liberated into underground air, gas is distributed along ventilation paths in difficult to predict ways and time duration. Gas might be observed on unlikely for common understanding places. By that reason preliminary knowledge about gas dynamics is extremely valuable. One possible approach might be real mine experiments. This approach however faced great difficulties due to problems with simulation of realistic gas liberation characteristics. Problems with control of such simulation cannot be neglected as well. All the above points show that mathematical and computer

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Finite volumes for complex applications

modeling can be of favor in prediction of gas distribution along ventilation paths. They can reveal specific relationship between gas liberation and its influence on underground air flows.difficulties with control and safety of the experiment. In that connection mathematical and computer modeling in close with reality decsription is very important and liable technique. It can reveal specific relationships between gas source and its influence on ventilation system and its important parameters. The process of gas liberation and distribution in a single path and its mixing with fresh air is presented by convection-diffusion equation with source function, varaible velocity and density of air gas mixture. It is solved simultaneously with continuity equation. Control volumes method with exponential scheme is applied for numerical modeling.. Errors due to discterization and limits of method for this class of problems are outlined. Solutions for one ventilation path are addapted to a network modeling where interrelation between air flows with different characteristics are of a great importance to the whole process of safety working conditions in mine. Results of modeling are validated with practice in the following ways: ^ it was used as investigation tool for one methane explosion which took place two years ago in Bulgaria. Revealing of most likely reasons for its occurrence became possible; > it was also applied to in one case study for inertization with nitrogen of a typical mine configuration. This technical activity is performed to supress mine fire development. 1.

Governing equations and source functions

Let us assume mine roadway with characteristics shown on figure 1. Air flow is well developed in direction of coordinate s. Length of roadway (L) is times greater compared to its width or cross sectional area (f) and this give a ground to present mathematical model into one dimensional way. Independent variables are: • s - along the length of mine roadway, m; • 1 - time for tracing the process, s.

Figure 1. Physical model.

Figure 3. Discretization of calculation area

Known functions are: U(S,T) - velocity of air gas mixture, m/s;

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q(s,i) - gas liberation, m3/s; P(S,T) - density of air gas mixture, kg/m3. Unknown function is C(S,T) - concentration of gas impurity into air gas mixture. Gas distribution in thus describe physical model obeys mass conservation for gas impurity and air and equation of state for air gas mixture: • •

In the above written equations index g referred to gas impurity, index m - to air gas mixture. Initial and boundary conditions are given with the terms:

Source functions q(s,T) which present gas liberation or provocation of transient process are presented on figure 2. Their characteristics may change either in time or in space or combinations of both. They are chosen on typical behavior of gas sources: a - gas liberation from walls or during coal transportation; b, c - gas liberation from already mined areas; d - point gas source; e - gas liberation due to repeatable operations (diesel power, blast work); f - gas liberation from mined areas with increase of its surface; g - gas desorption from newly opened surfaces. Figure 2. functions

Typical

source

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Finite volumes for complex applications

Combinations of above presented source functions are also possible. For istance: a + d - point gas source with constant emission rate in time; c + f - distributed along the length source with variable in time emission rate; b + g - linearly distributed source along the length with pick in time and then gradually decreasing emission rate; d + e - point source with periodically changeable emission rate. Presented on figure 2 source functions can be easily approximated with polinom or with sum of functions. Numerical treatment of above presented mathematical model - expressions (1-3) plus (4-5) makes possible taking into account all written into model variations. 2.

Numerical and computer modeling

2.1. Schemes, mesh-type, discretization Mathematical model (1-5) is solved by application of Control Volumes Method [PAT84]. Presented on fig. 1 physical model is transformed into regular mesh of control volumes (fig. 2) with size As. They are defined by mesh points and control volumes boundaries. Number of points are Concentration

C(Si,Tj)

is

defined

in

mesh

points,

air-gas

flows

Exponential profile [PAT84} for concentration variations is assumed. Following the above points numerical analogue, binding three adjacent points (W,P,E) can be written in the way:

where:

Upper index 0 referred to a previous time step. Boundary conditions (4-5) are transformed into numerical schemes in the same way by assuming first order boundary condition at left boundary (s=0) and second order - at the right boundary.

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Full procedure on transformation of differential problem into numerical schemes is given in [VLA93]. Having expressions (7) for each three adjacent points linear system with three diagonal matrix for all points i d ( l - N x ) i s obtained. Its solution under TDMA gives concentration in each mesh point and in any time moment C(SJ,TJ). 2.2. Approximation, stability, convergence, errors, application limits Numerical schemes (7) approximate differential problem (1-5). This statement was proved by application of Taylor's series. Approximation error is from first order in regard of 1 and from second order in regard of s. Numerical scheme (7) is absolutely stable. It was proved by applying Matrix criterion [SHI88] by presenting numerical scheme in way from one time layer to the next one and examining eigen vectors and their eigen values [SHI88]. Convergence is a concequence from approximation and stability, following Lax theorem [SHI88]. Applicability limits to gas dynamical problems are investigated. For most common parameters - velocities, lengths etc. error analysis was performed. On figure 4 relative errors (numerical/exact solution) in regard of

^

Cu number (Cu =

Figure 4. Relative errors due to discretization

3.

uAT

As

). Number of

mesh points, even limited to the minimal range (3) lead to 8% errors, which for purpose of gas dynamical problems is agreeable. Physically unlikely results has not been obtained.

Modeling validation with practice

Presented herewith mathematical model and its numerical interpretation reach their computer realization in both aspects: • for a single roadway or set of sequential ones [VLA93]; • for a complex network, where solutions for a single roadway were harmonized at places where they cross each other [STE87]. Modeling was applied in numerous engineering problems from practice. Two of them are presented in this paper: • investigation of one methane explosion; • technological solutions for inertization with nitrogen of fire zone. 3.1. Methane explosion

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Severe methane explosion took place on the 1-st of September 1997 in "Ivan Russev" coal mine. Operations in one section were canceled due to annual holidays from 1st August to 1st September. As a result great amounts of methane were accumulated. Degazation (transfer of this gas out of mine) was to be performed. Mine does not have remote control system and no data (besides miners' evidences) was available. As a result of incorrect activities of miners, performing degazation, explosion took place causing death for more than 10 workers.

Figure 5. Path in mine for distribution of methane during degasing Investigation about circumstances [MIC98] was performed by application of models, presented in this paper. Investigated sector of mine is shown on figure 5.

Figure 7. Concetration at the entrance of Figure 6. Air volumes at the entrance of observed path observed path Changes in cross sections can be seen there. Transient process was initiated by variable air flow (figure 6) and methane concentration (figure 7) at starting point 0. They reflect workers actions, such as: • variable cross-sections along the route (fig. 11); • assembling of ventilation curtain to direct higher air flow towards the gassed section; • switching on/off of booster's fan operation, causing changes in ventilation conditions (air volumes Q 1 Q 2

and

Q3

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variable methane concentration (C2) of in-flowing air into 503 crosscut (Q2)

Explosion took place between points 5 and 6 and modeling results (figure 8) show the same - high methane concentration with more than 7 minutes duration stay at the place of explosion. Unfortunately ignition source was also available. Figure 8. Concetration in time of observation in some points of path

3.2. Inerting with nitrogen of fire zone In some cases during mine operations evidence for [MIC97] development of mine fire can be observed. Then one of anti-fire measures is to inertize atmosphere at a danger zone so that oxygen does not be available to support burning. The model presented herewith can deal with one impurity into air-gas mixture. Inerting of air however presumes more than one ipurity, namely: • methane inflow from mined zone (points 5-6 on figure 9) and from mine workings (points 1-2-3-4 on figure 9); • oxygen from ventilation flow and from injected technical nitrogen; • nitrogen from air and from injected technical nitrogen. In order to evaluate concentration of the above mentioned three gas components via a model constructed for a single component, the author has applied consecutive diffusion mixing. It pressumes appropriate definition of transporting medium and impurity as well as suitable presentation of gas sources q(s, T) . Calculational passes three stages (methane release and distribution, nitrogen outflow from gob area, nitrogen injection at a given place in the panel and its further distribution in the area which must be inert). For any of these stages computer modulus were developed - METHANE, INERT_GOB and NITRO. Common input data for the three computer programs are geometric characteristics of mine workings (fig.9). The three programs interact and their incorporating in the total inverting strategy makes it possible the composition of general program procedure INERTIZATION Numerous solutions were performed, corresponding to specific fire situations. Very important parameters were obtained: • time needed to inert the observed object; concentration of flammable gas in all points of observed area (this Figure 9. Object for inertization

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information is very important in order to keep the atmosphere out of explosion); • effectiveness of operations performed on fire supression. On figure 10 is shown methane concentration along the path with length 1500 m. Figure 11 shows time delay in inertization from point 2 and 3.

Figure 10. Time delay in inertization

Figure 10. Methane concentration

4. Bibliography [VLA93]

[MIC98]

[MIC97]

[PAT84] [SHIS8] [STE87}

Vlasseva E.D., Mathematical Modeling of Convection-diffusion Processes in Underground Mines, Ph.D. Thesis, UMG, Sofia, 1993, pp. 200 Michaylov M.A., E.D.Vlasseva, Simulation Analysis of Methane Explosion, Second International Symposium on Mine Environmental Engineering, July 29-31 1998, Brunei University, UK,p. 1-16. Michaylov M.A., E.D.Vlasseva, Modeling of Preventive Nitrogen Inertization in Underground Mines, 15th Mining Congress of Turkey, 6-9 May 1997, Ankara, pp. 203-210. Patankar S., Numerical Methods in Heat Transfer, Sovremennoe Mashinostroene, Moskow, 1984, p. 150 (Russian translation) Shi D., Numerical methods in Het Transfer Problems, Moskow, Mir, 1988, p. 544 (Russian translation) Stefanov,T.P., E.D.Vlasseva, E.E.Arsenyan, Unsteady Gas Flows in Mine Ventilation Networks, 22 International Conference of Safety in Mine Research Institutes, Beijing, China, oct.1987,pp. 115-124.

Simulation of salt-fresh water interface in coastal aquifers using a finite volume scheme on unstructured meshes B. Bouzouf, D. Ouazar LASH, EMI 14 Av. Ibnsina Rabat, Maroc I. Elmahi IVG, University of Duisburg, Germany

ABSTRACT This paper is devoted to the numerical study of seawater intrusion into coastal aquifers. The cell-centered finite volume method is adopted here to solve the set of simultaneous partial differential equations describing the motion of saltwater and freshwater separated by a sharp interface. These equations are based on the Dupuit approximation and are obtained from integration over the vertical dimension. In order to have flexibility upon complex configurations domain, non structured grid meshing is utilized. To approximate the diffusion fluxes, GreenGauss type reconstruction, based on Diamond cell and least square interpolation, is performed. The model is first validated using academic test case studies with known close from solutions. A real case study concerning the Gharb aquifer in North West of Morocco is carried out to show the overall trend of saltwater fronts. Key Words: Coastal aquifers, seawater intrusion, finite volumes, unstructured meshes, Green-Gauss reconstruction. 1. Introduction The modelling of groundwater in coastal aquifers is an important and difficult issue in water resources. The primary difficulty resides in efficient and accurate simulation of the movement of the saltwater front. Freshwater and saltwater are miscible fluids and therefore the zone separating them takes the form of a transition zone caused by hydrodynamic dispersion. For certain problems where the transition zone is relatively small compared to the aquifer extent and thickness, the simulation can be simplified by assuming that two fluids are immiscible and separated by a sharp interface (sharp interface model). This last assumption, together with the Dupuit approximation, permits the integration of the equations in the vertical direction [BEA 79]. The objective of this paper is to present a cell-centered finite volume based approximation to calculate the position of the sharp interface. This class of methods

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is becoming one of the commonly used techniques for partial differential equations in engineering calculations and computational physics. Their popularity is due mainly to their ability to faithful to the physics conservation and the possibility of solving the problems on complex geometries. The diffusion contributions here are approximated by using Green-Gauss type interpolation. This technique is found to be very robust, it can be used on general mesh, not satisfying necessarily Delaunay condition on the triangulation. Time integration is performed by an explicit Euler scheme in order to keep the memory requirement reasonable. 2. Mathematical Model

We assume here that the saltwater and freshwater are separated by a sharp interface, thus two domains are considered. For each flow domain the equation of continuity may be integrated over the vertical dimension reducing the determination of the position of the interface to a 2D problem. The system of equations can be written as follow ([ESS 90]):

Qf and ns represent here the fresh and salt water flow domain respectively, Kfx and KSx (respectively Kfy and Ksy ) are the hydraulic permeabilities in the fresh and salt water in x-direction (respectively in y-direction), hf and hs are the heads, Bf and Bs are the thickness of fresh and salt water zone and n is the porosity. We note also by 6 = —, where pf and ps are the specific weights in fresh Ps - Pf and salt water, and by 0 for confined aquifer 1 for unconfined aquifer

(

Invoking continuity of the pressure at the interfaces, the interface elevation can be calculated from the freshwater and saltwater heads by

The system (1) represents two coupled parabolic partial differential equations which should be solved simultaneously for the freshwater head (hf) saltwater head (hs). Once these values are known, the interface elevation (£) can be obtained from (2). The set of the equations (1) can be written in the vectorial form:

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3. Finite volume discretization

To solve the system of equations (3) we have considered a triangular cellcentered finite volume formulation ([EBVGPH 99]), where the state variables W™ are the average values for the cells at time level n:

Integrating eq. (3) on a control volume d yields in explicit formulation:

where Ai is the time step. The discretization of equation (4) requires the approximation of terms such as

where / = /, s and F^ is the interface separating two cells d and Cj. Y. Coudiere et al. [CVV 96], have studied an elliptic problem

Where A is a symmetric definite positive matrix with coefficients aij in C1(il), / € C°(fy and g € C2(T). They have used a Green-Gauss type interpolation to construct the gradients at the interfaces of the mesh. The gradient on each edge is approached by the Green theorem and then a first order Gauss quadrature formula, for which requisite values at the vertices P are interpolated from the states on the neighbourhood of P. The weak consistency of this scheme was

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proved under some assumption on the weights of the interpolation. We took inspiration from this scheme for devising our numerical procedure and discretize the diffusive contributions. We begin firstly by writing

One constructs the co-volume Cdec centered at the interface Fij and connecting the barycenters Gi and Gj of the triangles that share this edge and the two endpoints N and S of FJJ (see figure 1).

Figure 1: Diamond shaped co-volume or

To calculate -^— | r • • , the divergence theorem is applied to the co-volume Cde ox surrounding Fij, which gives

e represents an edge of the co-volume Cdec and nxe is the axial component of the outward unit normal to e. If we note by e — [N1, N2], One can write also

Where hi N1 and hi N2 are respectively the values of hi on the nodes N1 and N2 of the edge e. The data at the centers d and Gj are known exactly while the data at the vertices N and 5 must be determined by some interpolation procedure. For one node P

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of the mesh, one utilizes a linear approximation v of hi on the set of cells which share the vertex P,

Where V(P) is the set of triangles K surrounding P, hiK the head at the center of triangle K and CCK(P) are the weights of the interpolation corresponding to the node P. In order to ensure the consistency of the scheme, the weights of the interpolation are calculated using a least square procedure (see [CVV 96] for details). 4. Model validation

To verify and validate the numerical solution obtained from the finite volume model, numerical simulations have been compared to existing analytical solutions. 4.1 Steady state Two cases have been checked: confined and unconfined aquifer. For both of them the initial values of hf and £ are arbitrarily fixed. The analytical solutions are as follows: . Unconfined aquifer ([VER 68], fVN 751):

with

and

• Confined aquifer ([GLO 59},{RH 62]):

with 4.2 Unsteady state Keulegan [KEU 54] presented an analytical solution for the interface in a confined aquifer of uniform thickness:

with D = 10m , n = 0.3 and K = 39.024m/day.

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Figure 2: Comparison of analytical and numerical solutions for unconfined aquifer.

Figure 3: Comparison of analytical and numerical solutions for confined aquifer.

Figure 4: Comparison between analytical solution and finite volume method.

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The numerical solutions are in good agreement with the analytical solutions as depicted in figures (2), (3) and (4). 5. Application to the Gharb basin, Morocco

We have applied the finite volume based model to the Gharb aquifer which is located in North West of Morocco. The surface area of the coastal Gharb is estimated to about 4000 km 2 . It belongs to the structural domain of the Morocco Atlantic plain.

Location of the Gharb basin In figure (5), the areas of pumping in the Gharb aquifer are depicted. Figure (6) shows the actual front saltwater corresponding to pumping schemes of Figure (5). 6. Conclusion

Characterization of certain coastal aquifer systems may be accomplished by assuming that saltwater and freshwater are separated by a sharp interface. Invoking the Dupuit assumption and performing a vertical integration results in quasi-three-dimensional, the equations may be solved to give freshwater head, saltwater head and interface elevation. Cell-centered finite volume scheme on a unstructured mesh is used to approximate the partial differential equations. Comparisons of the finite volume approach adopted in this paper, with known analytical solutions have shown close agreement. The model was also applied to a real case concerning the Gharb aquifer in North West of Morocco.

BIBLIOGRAPHY [BEA 79]

BEAR J., Hydraulics of groundwater, McGraws-Hill, New York, 569 pages, 1979.

[CVV 96]

COUDIERE Y., VILA J. P. AND VILLEDIEU P., Convergence of a finite volume scheme for a diffusion problem, F. Benkhaldoun

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Fig. 5 Areas of pumping in GHARB Fig. 6 Simulated freshwater and saltwater aquifer interface

and R. Vilsmeier eds, Finite volume for complex applications (Hermes, Paris), pp. 161-168, 1996. [EBVGPH 99]

ELMAHI I., BENKHALDOUN F., VILSMEIER R., GLOTH O., PATSCHULL A. AND HANEL D., Finite volume simulation of a droplet flame ignition on unstructured meshes, J. of Comput. and Appl. Math., Vol 103, 1, pp. 187-205, 1999.

[ESS 90]

ESSAID H. I., A quasi-three-dimensional finite difference model to simulate freshwater and saltwater flow in layered coastal aquifer systems, U.S. Geological survey Water-Resources Investigations, Report 90-4130. Menlo Park, California, 1990.

[GLO 59]

GLOVER R. E., The pattern of freshwater flow in a coastal aquifer, J. of Ground Water Resour., 64, pp. 439-475, 1959.

[KEU 54]

KEULEGAN H. G., An example report on model laws for density current, U.S. Natl. Bur. of Stand., Gaitherburg, Md, 1954.

[RH 62]

RUMMER R. R. AND HARLEMAN D. R., Intruded saltwater wedge in porous media, U.S. Geol. Surf. Prof., Paper 450-B, 1962.

[VER 68]

VERUIJT A., A note on the Ghiben-Herzberg formula, IASH bull. 13, pp. 43-45, 1968.

[VN 75]

VAPPICHA V. N. AND NAGARAJA S. H., Steady state interface in coastal aquifer with a vertical outflow face, National Symposium on Hydrology, Rurkee, India, 1975.

Progress in the flow simulation of high voltage circuit breakers X. Ye, L. Miiller, K. Kaltenegger and J. Stechbarth ABB High Voltage Technologies Ltd., 5401 Baden, Switzerland ABSTRACT In this paper progresses in the physical and numerical modelling, which lead to improvement in the accuracy and capability of simulation for capacitive switching design of circuit breakers, are introduced. Numerical results and measurement results are compared and discussed. One important progress lies in the successful treatment of the artificial viscosity. To maintain the numerical stability but at the same time to keep the artificial viscosity so small that the physical viscosity is not distorted, the upwind biasing essentially local extremum diminishing (ELED) scheme has been adopted and improved. Another progress is associated the moving grids technique, where additional terms have been added to the governing equations for the moving grid without deformation and new grid lines are added or removed in the deformed moving grid by the solver in the progress of the calculation. The Chimera boundary interpolation method is used to enable the communication between two blocks with relative motion. Key Words: circuit breaker, moving grids, upwind biasing scheme, Chimera boundary

\. Introduction In a high voltage circuit breaker a gas with good dielectric and thermodynamic properties such as SF6 is used to extinguish the electric arcing which occurs as electric contacts move apart. The current interruption can only then be realised. The capacitive switching, i.e., the current interruption at high voltage but with low current, represents one important case among various cases of tests and design. The ability to perform capacitive switching without electric breakdown is one of the defining parameters for the speed of the circuit breaker and therefore strongly cost relevant. Furthermore, only the density of gas (p) and its electric field strength (£) play an important role in gas breakdown between the contacts of the circuit breaker (s. Fig. 4). As the criterion of the ratio ofE/p for gas breakdown is well known, CFD can be employed to simulate the flow field and to subsequently produce the distribution of gas density in the circuit breaker during the design. It is obvious that for such simulations, a high degree of accuracy of flow calculation is required.

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Since later 1980's researchers and engineers have begun to apply CFD tools in the development of circuit breakers. Most of them, e.g. [CLA 97], concentrated their efforts on using the flow simulation with implemented arcing models to investigate the ground flow effects in circuit breakers, e.g., pressure build-up in pressure chamber, plasma jets, and to identify the limit and capability of CFD tools. Their efforts to verify CFD tools were, however, constrained mainly in the comparison of the simulated pressure build-up in the pressure chamber with experiments. There are only few works (s. [TRE 91]) done for enhancing the accuracy of CFD tools for predicting dielectric strength in capacitive switching and for verification of CFD tools in the regions of nozzles and electrode contacts. As a result, the ability and accuracy to predict the dielectric strength of a circuit breaker with CFD tools have been not satisfactory. The following two points remain unclear. 1) How can the moving parts be treated correctly and which numerical methods are to be introduce to obtain a sufficient high accuracy? 2) How can a CFD tool be verified for predicting dielectric strength in capacitive switching? In this paper, these questions are addressed, with concentration on the simulation of cold SF6 gas with moving electric contacts and further on the prediction of the dielectric strength of a circuit breaker during the capacitive switching. The cold gas simulation is not as trivial as to be anticipated, because the low energy level of cold gas forces a CFD code to consider, with sufficient accuracy, all macroscopic and microscopic flow effects, examples of which are: flow separation, influence of wall and viscose layer and their transient evolution, turbulence transport and suck effect caused by moving contacts. Further, to guarantee the numerical stability, artificial viscosity must be introduced, and hence a sophisticated scheme must be used to keep the artificial viscosity so small that it can maintain the numerical stability effectively but does not confuse the physical viscosity on a viscose layer. A sophisticated moving grids technique has to be introduced both for obtaining a high accuracy and for performing an efficient computation. In section 2, the mathematical models and numerical methods, such as ELED scheme, moving grids technique and Chimera boundary, are introduced. In section 3, examples are presented and discussed for flow simulation where the code verified through the comparison with experiment. Subsequently, the method and example are introduced for coupling of the flow field and the electric-static field in circuit breakers with the consideration of the influence of roughness. Finally, in section 4, our results are concluded with further improvement suggested.

2. Mathematical models and numerical methods

2.1 Mathematical models For the fluid flow in a circuit breaker it is necessary to use the complete NavierStokes equations in their time averaged form, i.e. the so called Reynolds equations

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with an adequate turbulence model. The fluid flow will be then governed by the Navier-Stokes conservation equations for mass, momentum and energy. These equations have the differential form as shown in eq. [1].

with the variables

with the shear stress I and heat flux Jas follows:

where ///, //, are molecular and turbulent viscosity, A heat conduction coefficient, 0 temperature, Pr Prandtl number, V Nabla operator, Re Reynolds number, W velocity vector. These equations can be transformed to curvilinear coordinate system with the transformation <^=^(x,y) and T]=T](x,y). To close the equation system, the standard k-£ turbulence model (s. [LAU 74]) was used to obtain the turbulent viscosity ju,.

2.2 Numerical methods The basic equations are solved with a multi-block finite volume Runge-Kutta multisteps time-marching method. The code (HT206) was previously applied for fluid flow in turbo systems (s. [SCH 91] and [SCH 98]) and has been extended for the development of circuit breaker. The numerical methods of the code, which are relevant to the flow simulation of circuit breaker, are described as follows: 2.2.1 ELED scheme The spatial discretisation of the present code is based on a high order non-oscillatory scheme, which consisting of central discretisation and artificial viscosity in the following form (s. [JAM 81]):

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Finite volumes for complex applications £{ '

where q denotes the variables in the equation [1] and ,.i and "y+i are the ;4 coefficients of the scheme. If the formulation of these coefficients is taken from [JAM 81], it is then the classic Jameson-Schmidt-Turkel (1ST) scheme. A more advanced formulation of these coefficients is introduced in the present code, that is the essentially local extremum diminishing (ELED) scheme of [JAM 94]:

The valuation of the wave speed J+- in £ and 77 direction here will be treated generally for variable q overall in the flow field as

where/and g are convection terms in £ and 77 direction, U and V are defined in eq. [3] and [4]. Based on our experience the numerical constants r, £, C, and C2 can be selected as r=1.5, e=10-10, Cy=2.0, C2=1.5 - 8.0. This formulation is defined to be scheme which satisfies the condition that in the limit as the mesh width Ax —> 0, local maxima are non-increasing, and local minima are non-decreasing.

2.2.2 Moving grids technique To simulate the flow field containing electrical contacts moving during the current separation, two types of moving grids are generally used. The type I moving grid moves only its form and the number of grid will be not changed as shown in Fig. la. In contrast with the type I, the type II moving grid will be expanded or compressed by adding new grid lines to or removing existing ones from left or/and right side as shown in Fig. 1b. Some methods have to introduced to treat both types of moving grids correctly.

Figure 1. Two types of moving grids

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2.2.2.1 Treatment of moving grid type I In the type I moving grid, the variables will be kept in the same cells during the movement at a time step. This will produce error if no additional measures will be taken, because the positions of these values in the flow field have been changed through the moving of the grid cells. To avoid such errors, additional terms in eq. [1] must be introduced. In the case that a block of grid moves with a speed of XT and yT as its x and y components, the velocities U and V in the curvilinear coordinate system with £ = %(x,y,t) and 77 = Tf(x,y,t) will then be (s. [STE 78]):

where <^T = -x^x -y£y and %r = -x£x -y£y, then the convection term in eq. [1] after the transformation to the curvilinear coordinate system is

2.2.2.2 Treatment of moving grid type II For the type II moving grid, the movement only involves the two side grid lines, i.e., almost all the cells of this grid are not in motion, therefore, there is no need to apply the additional terms. However, because of the change in the cell size at every time step, there would be implicit unphysical energy and mass loss as grid compresses and unphysical energy and mass increase as the grid expands, if the values of energy and mass after the change of cell size were taken simply from the old before the change of cell size. Therefore, sophisticated treatment of the value of energy and mass is very important to get the accuracy of calculation. As grid expands or compresses, the value of the density and the total energy in the changing cell after the change of cell size will be then

where Xj.j-Xi is the cell size, new and old denote the values after and before the change of cell size.

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Finite volumes for complex applications

2.2.3 Chimera boundary Because of the movement relative to neighbouring grids, the moving grids will have discontinuity of grid line at interfaces with their neighbouring grids. Hence, the Chimera interpolation method (s. [STE 87]) will be applied to enable the information exchange between two grids with discontinuity of grid lines. The basic concept is that two layers of ghost cell of a block will overlap with the neighbouring block. To obtain values in the ghost cell, the values of the nearest cell centres of the neighbouring block are interpolated to the cell centre of the ghost cell (s. Fig. 2). The values in the ghost cells can then be used for flux building. Same as the central discretisation, the Chimera interpolation is non-conservative, however, the upwind character is obtained through the ELED scheme.

Figure 2. Overlapping blocks for Chimera interpolation 3. Results and Discussion 3.1 Improvement of numerical scheme To investigate the improvement of the numerical scheme especially in its capability to resolve the flow discontinuity and to keep a small disturb in shear layer, two test cases are considered.

Figure 3. Two test cases for numerical schemes

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The first test case is the shock, tube problem as described in [SCH 98]. In Fig. 3a, the pressure distributions calculated with the JST and ELED schemes are compared with the exact solution of [HIR 90] at the time t=6.1 ms. As can be seen, the scheme of JST oscillates strongly just before the shock front whereas the ELED scheme resolves the discontinuity without oscillation. The use of ELED scheme resulted in only 7 percent increase in the computing time. The other test case is the fully developed turbulent pipe flow as shown in Fig. 3b, where the calculated fully developed radial velocity distributions, which is normalised by the velocity in the middle of pipe Um, are compared with the experiment results of [NIK 32]. It can be seen that the near-wall shear layer is strongly disturbed by the JST scheme, while the result of ELED scheme agrees very well with the experiment.

3.2 Verification with experiments of a circuit breaker Fig. 4 shows a schematic diagram of the core part of a circuit breaker. To interrupt the current, the electrical contact "finger" will be moved together with isolating and auxiliary nozzles toward left, while the electrical contact "plug" will stay still. The gas in the pressure chamber (not illustrated) will be then compressed through this motion and flows from left into the isolating nozzle, there will be a highly transient and transonic flow.

Figure 4. Schematic diagram of the simulated circuit breaker To verify the code for the capacitive switching design, measurement of static pressure on the 6 points showed in the Fig. 4 was carried out for cold. These 6 measured points are located in different flow regions. ® and (D are in the diffuser region with flow separations; ® is in the geometric throat; ® and © are in the channel flow region with boundary layer character; (D is behind the shock front and presents the pressure lost over the shock. Therefore, the values at these 6 positions reflect all flow details. For the calculation, an inlet was defined as shown in Fig. 4 and the pressure measured in the pressure chamber was used as the inlet boundary condition, so that the leakage in the pressure chamber which is difficultly to be estimated can be ignored. In Fig. 5 the simulated pressure distributions at © to ® are

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Figure 5. Comparison of pressure in circuit breaker: simulation, O measurement

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compared with the measurement. The simulation results agree very well with the measurements. The discrepancy of P5 is caused by the selection of the inlet where the boundary layer begins downwards from the start point in the reality.

3.3 Coupling of the electrical and flow fields Based on the test results on a circuit breaker where breakdown voltages were measured in dependence of travel positions, the present code can be verified through comparison of the ratio E/(pkr) with the critical value of 1480 kVm 2 /kg for 6 bar absolute filling pressure of SF6 from the streamer theory (s. [BEY 86]). In this case E/(pkr) must be higher than the critical value, where kr is a roughness factor which accounts for the microscopic effects which intensify the local field strength and is obtained from an semi-empirical function of local density and roughness. The electrical field strength E was calculated with the ABB electrical field program ACE. The gas density p was obtained from the flow simulation with the present code. Fig. 6 shows the distribution of E/(pkr) in the circuit breaker. The maximum of this ratio lies on the surface of the plug contact and it overruns the critical value slightly. The simulation results correspond very well with measured breakdown/hold values of the applied voltage observed in measurements.

Figure 6. E/(pkr) distribution and its maximum at travel=l 15 mm

4. Conclusion The progresses in the numerical methods, including numerical scheme and moving grid technique, have been made, leading to the development of the code presented in this paper which is able to fulfil the requirements of capacitive design of high voltage circuit breaker. The effectiveness of the code is confirmed through the following facts: (1) The adoption of the more advanced ELED scheme resolves flow discontinuity efficiently; (2) the physical viscosity is not confused; (3) the calculation results agree very well with the measured results; (4) the calculated

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density was coupled with electrical field strength, and the resulted ration of E/(p kr) predicted the gas breakdown correctly. The code will be further improved and developed, in particular, arcing model will be implemented and verified, so that it can be used to predict the dielectric strength under high temperature and pressure.

5. References [BEY 86] Beyer, M, Boeck, W., Moller, K., Zaengl, W.: Hochspannungstechnik, SpringerVerlag, Berlin Heidelberg New York, 1986 [CLA 97] Claessens. M., Moller, K., Thiel, H.G.: A computational fluid dynamics simulation of high- and low-current arcs in self-blast circuit breakers, J. Phys. D: Appl. Phys. 30, p. 2899-2907, 1997 [HIR 90]

Hirsch, C.: Numerical computation of internal and external flows, Vol. 2, John Wiley & Sons, 1990

[JAM 81] Jameson, A., Schmidt, W., Turkel, E.: Numerical solutions of the Euler equations by finite volume methods with Runge-Kutta time stepping schemes, AIAA paper 81-1259, January, 1981 [JAM 94] Jameson, A.: Analysis and design of numerical schemes for gas dynamics 1: Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence, Int. J. of Computational Fluid Dynamics, August, 1994 [LAU 74] Launder, B.E., Spalding, D.B.: The numerical computation of turbulent flows, Computer Methods in Applied Mechanics and Engineering, Vol. 3, p. 269-289, 1974 [NIK 32]

Nikuradse, J.: Gesetzmafiigkeit der turbulenten Stromung in glatten Rohren. Forsch. Arb. Ing.-Wes. Heft 356, 1932

[SCH91] Schafer, O.: Application of a Navier Stokes Analysis to turbomachinery bladecascade flows, 19th International Congress on Combustion Engines, CIMAC, Florence, 1991 [SCH 98] Schafer, O. et al: Last advances in numerical simulation of aerodynamic forces on turbine blades of turbochargers for pulse charged engines, 22nd CIMAC , International Congress on Combustion Engines, 19-21 May , Kopenhagen , 1998 [STE 78] Steger, J.: Implicit finite-Difference simulation of flow about arbitrary twodimensional geometries, AIAA Journal, Vol. 16, No. 7, July, 1978 [STE 87]

Steger, J., Benek, J.: On the use of composite grid scheme in computational aerodynamics, Computational Methods in Applied Mechanics and Engineering, Vol. 64, No. 1-3, 1987

[TRE 91] Trepanier, J.Y. et al: Analysis of the dielectric strength of an SF6 circuit breaker, IEEE Transaction on Power Delivery, Vol. 6, No. 2, April, 1991

River valley flooding simulation

Francisco Alcrudo Area de Mecdnica de Fluidos Maria de Luna, 3 CPS - Universidad de Zaragoza 50015 Zaragoza, SPAIN

ABSTRACT

Dam break flood wave propagation along a reach of a river valley located in the Italian Alps is mathematically modeled with package SW2D that solves the nonlinear Shallow Water equations. Simulation results are compared with data obtained from a physical model of the river valley operated by ENEL (Italy). The difficulties encountered during the modelisation process and the solutions adopted are explained in this paper. Key Words: Dam break, Flood, Shallow Water, Physical Model

1. Introduction Considerable efforts are being presently devoted to the validation of numerical models describing dam break flows, mainly due to the need for modern risk assesment and mitigation tools. Real life experimental data concerning actual dam break or severe flooding are very difficult to obtain because of the unpredictable nature of the phenomenon. However, measurements obtained from reduced scale physical models can provide excellent validation information because the experimental conditions can be more precisely defined. The work reported in this paper concerns the comparison of the simulation results obtained with SW2D program [ALC92] with measurements of the flooding experiments carried out by ENEL (Italian Utility Company) in a reduced scale physical model of the Toce river. The physical model is some 50m long by llm wide and is built mainly in concrete (see figure 1). It represents a 5km long reach of Toce river which is located in the Italian Alps.

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Figure 1: Digital Terrain image of Toce river valley physical model (ENEL) The model reproduces many details of the actual valley geometry including the river bed and some villages, hydraulic structures and a reservoir located in the middle of the reach that depending on the intensity of the flood is overtopped and eventually filled with water across its embankments. The upstream end of the river reach model is connected to a small water tank fed by a hydraulic pump. Flooding is initiated by starting the pump that rapidly fills up the tank, overtops the entrance to the reach and rushes downstream. The pump capacity is such that the process takes place very rapidly thus simulating an abrupt irruption of water into the valley model. ENEL personnel located water stage probes at 32 different positions in the model valley. Among them one in the river bed at the entrance section and another one in the middle of the feeding tank that can be used to impose the boundary conditions together with the pump discharge versus time that was also recorded. Experiments were carried out for two flooding intensities: The first one such that no overtopping of the reservoir takes place (peak discharge of 0.21 m 3 /s) and the second one with reservoir overtopping (peak discharge of 0.36m 3 /s). Measured water stage readings at several probe locations were compared to water levels obtained with SW2D model for different friction coefficients and flood intensities and overall satisfactory agreement was found. The physical model geometry was distributed by ENEL as a Digital Terrain Model (DTM) covering the model area at regular intervals of 5cm, therefore specifying the bottom elevation function z s ( x ^ y } in some two hundred thousand points. The simulations reported here were run on a platform of comparable computing power to that of a Pentium processor. In order to have reasonable run times (a few hours) the size of the DTM grid had to be coarsened by a factor of three.

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2. Mathematical model It is commonly accepted that bulk flood flow can reasonably be well described by the non linear Shallow Water equations that simply express conservation of mass and momentum in the plane of movement of water. Since they are obtained after and integral mass and momentum balance in the horizontal directions (or by averaging the Navier-Stokes equations across the vertical) no information regarding vertical velocities is obtained. Usually no flow shear forces are taken into account when the problem is convection dominated as it is the case in severe flooding. Friction forces with the bottom are accounted for by empirical formulae such as Manning's or Chezy's. The Shallow Water equations can be written in integral conservation vector form as follows:

Here t represents time, dV an elementary volume and nx and ny the cartesian components of the normal vector to the elementary surface area dS enclosing the considered volume. Think that in 2-D a volume means in fact an area and an area is actually a line. U is the vector of conserved variables and F and G are the cartesian vector fluxes of mass and momentum.

Here h, u and v represent water depth and the two cartesian velocity components respectively and g is the acceleration of gravity. Source term H accounts for bed friction and bottom slope:

where n is Manning's friction coefficient. The mathematical model SW2D solves the Shallow Water equations in two dimensions by means of a finite volume spatial discretization in multiblock structured meshes coupled to an explicit two step time integration scheme. This is done by applying equation (1) to every cell of the computational domain in the usual Finite Volume approach. Cells can be quadrangles of arbitrary shape but sound judgement has to be exercised so as to avoid very distorted or stretched control volumes that may degrade the overall accuracy. Numerical fluxes are evaluated at cell faces through MUSCL variable extrapolation with limiting to enforce monotonicity. After variable extrapolation, Roe Riemann solver is applied at each cell interface. Bottom slope and bed friction represented by Manning's formula are spatially integrated pointwise.

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Figure 2: Cartesian grid used in the computations totalling 22000 points Once the spatial discretisation has been done the solution is advanced in time by a predictor-corrector sequence. Source terms are implicitly time integrated with no extra cost because the operator remains diagonal if they are pointwise spatially discretised. Details of the algorithm can be better found in [ALC98]. 3. Boundary conditions Proper computation of the flow variables at the upstream end in order to reproduce the correct flood characteristics at the inflow of the model valley is crucial if good agreement with downstream located probes is sought. Downstream boundary conditions do not exert such a strong influence on the global flowfield mainly because water leaves the reach in critical or supercritical conditions. Available initial data from ENEL were the inflow rate, Q, the reading of the water level probe located in the inflow tank, named SI, and the reading of the water level probe located at the inlet section, named S2. After considerable efforts it was determined that good agreement with experimental data at the entrance could be found only by imposing a subcritical inflow condition based on the available total head at the feeding tank. Despite the advise given by ENEL that flow conditions are critical at the entrance section, the model could not be run under this assumption. Numerical experiments showed that failure to accept critical flow at inlet was due to a slight adverse slope in this area that led to flow reversals, because critical flow can only be reached at the top of an upslope. Since subcritical inlet conditions require that two flow variables be imposed, they can be implemented by either imposing flowrate, Q, water level at the inflow section (S2 probe reading) or far upstream on the reservoir (SI probe reading) together with the inlet angle, a.

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Imposing flowrate is not adequate in a two dimensional computation involving an irregular inlet section because it is difficult to obtain an appropriate criterion to distribute the available discharge among the inlet section cells. Using the water level at the inflow section is an interesting option but it is better to use the water level at the feeding tank (SI probe reading) as the Total Head available in order to have water level at the inlet section (S2 probe) as an accuracy check. It must be borne in mind that the inflow rate, Q, can also be used as a check. Due to the size of the tank with respect to the entrance to the reach, the velocity in the former can be considered negligible, and the reading given by probe SI is considered as the total head /IT that is available at the inflow section. This can be written as follows:

being (h)si the reading of probe SI at the considered time and the other variables with subindex inflow are evaluated at every cell of the inlet section. Also from the outgoing bicharacteristic (see [ALC92] one has: (11-11 + 2c)inflow = (u • n + 2c)Mcfc

(5)

where u is the velocity vector and n the locally outward pointing unit vector. Subindex bich corresponds to the expression transported by the outgoing bicharacteristic from the inside of the computational domain. Once water depth and modulus of the water velocity at every inlet section cell are determined from the above equations the two cartesian components can be computed if an inflow angle, a, is imposed. In the tests run a was varied from zero to a few degrees with no significant changes in the computed results. 4. Testcases Besides the two inflow hydrographs (of different intensities) tested by ENEL, several simulations were performed varying Manning's friction coefficient above and below the value of n=0.016 suggested by ENEL. Also and more importantly, runs were made both with and without the buildings composing the valley villages. The DTM geometry did not contain buildings, but these could be included by modifying the bottom surface function Z B ( x , y ) appropriately. However, due to the low resolution of the grid used, buildings are represented very roughly as figure 1 shows: Villages can be seen as groups of mushroom like sprouts. Their influence on the solution is nevertheless very substantial. For every run made, great attention was paid to matching the inflow rate supplied by the pump (Q) and the water level measured by the probe located in the river bed at the inlet section (S2 probe). This guaranteed that at least the inflow flood wave was close to the actual one. Figures 3 and 4 show the

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Figure 3: Inflow rate and S2 probe reading for the moderate flood event

Figure 4: Inflow rate and S2 probe reading for the severe flood event comparison between computed and measured inflow discharge (left) and water depths in centimeters (right) at probe S2 (located just at the inflow section) versus time for the two tested flood events. Computational results (crosses, plusses and circles) correspond to different model options as shown. Although agreement in both flow and water depth is quite good, runs with buildings follow better the experimental values. Figures 5 and 6 show the comparison between calculated and measured water levels at probes S4 and P8 for both flooding events. Both probes are located around the central part of the valley, S4 being about 5m and P8 some 16m downstream of the inlet section. Although runs without buildings show larger errors than those including buildings (especially at probe S4) the situation is reversed at other probes not shown here for lack of space. Finally figures 7 and 8 show the same comparison at probes named P13 and P21. Both lie close to the river bed. P13 is located in front of the central reservoir about 21m downstream of the inlet section while P21 is located some 7m further downstream. Overall agreement at the considered locations can be judged acceptable.

Fields of application

Figure 5: Comparison at S4 and P8 probes for moderate flood

Figure 6: Comparison at S4 and P8 probes for severe flood

Figure 7: Comparison at PI3 and P21 probes for moderate flood

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Figure 8: Comparison at P13 and P21 probes for severe flood 5. Concluding remarks Although discrepancies between measured and computed water levels can be important at certain probe locations, the mathematical model used provides a reasonably accurate description of the two flooding events considered. Due to the very valuable assistance that this kind of tool can provide in tasks such as land use and emergency planning or risk assesment studies, it seems worthwhile to carry out further validation and improvement work. 6. Acknowledgements The author would like to thank ENEL and especially Dr. G. Testa for providing the experimental data and clarifying many technical questions. Finantial support provided by the European Union under CADAM concerted action is also gratefully acknowledged. 7. References [ALC 92]

[NUJ 95]

[ALC 98]

ALCRUDO F., Esquemas de alta resolution para el estudio de flujos discontinues de superficie libre, Ph.D. Thesis, Universidad de Zaragoza, 1992 NUJIC M., Efficient Implementation of non-oscillatory schemes for the computation of free surface flows, Journal of Hydraulic Research, 33, No. 1, 1995, p. 101-111 ALCRUDO F., Dambreak flood simulation with structured grid algorithms , Proceedings of the 1st CADAM (Concerted Action on Dam Break Modelling) Meeting, (1998), Published by the EU, in press.

Modelling vehicular traffic flow on networks using macroscopic models J.P. Lebacque1, M.M. Khoshyaran2

1

CERMICS-ENPC. TASC. USA.

2

FRANCE, email: [email protected]

ABSTRACT: In this paper, we describe a macroscopic model for vehicular traffic flow, with several extensions, resulting in a flow model on a network. These extensions require the introduction of link boundary conditions, partial flow dynamics and intersection models. Some numerical schemes based on the Godunov scheme are proposed for the discretization of the model. Key Words: Godunov scheme, Traffic flow, LWR model, partial flows

1

Introduction

Macroscopic modelling of vehicular traffic flow goes back to the pioneering work of Lighthill and Whitham [LW 55] and Richards [Ri 56], which introduced the celebrated LWR (Lighthill Whitham Richards] model of traffic on an infinite track. This model relies on the continuum hypothesis, i.e. the asumption that vehicular traffic can be described by macroscopic variables, the density K(x, t), the flow Q(x,t), and the speed V(#,£), as functions of the position x and the time t. These variables are related by the following equations:

or simpler:

Qe and Ve represent the equilibrium flow-density resp. speed-density relationdef

ships ( Q e ( K , x ) = KVe(K,x}). Their aspect is the following:

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Of course, considering the continuum hypothesis, the above LWR model (1) should be considered as a phenomenological model, but it is usually accepted that it provides a reasonably good description of the dynamics of traffic flow at a space scale of a hundred meters and a time scale of 10 seconds. Actually, the LWR model (1), also refered to as the first order macroscopic traffic flow model, constitutes but one among several competing approaches to macroscopic traffic flow modelling (see [LL 99] for a general discussion). Other notable modelling approaches include the second order macroscopic traffic flow models (see [Le 95] for an overview and [Sc 88] for relations between second and first order models) and the kinetic traffic flow models [PH 71], [Ph 79], [He 97]. Neither experimentation nor theory has provided arguments strong enough to support one model unambiguously. There is also no real consensus concerning the exact functional form of the equilibrium relationships, but the shapes suggested in the above illustration are generally accepted, up to a few variations. In the sequel, we shall concentrate on the LWR model, which is simple, enjoys obvious physical meaning, and provides results generally in good agreement with measurements.

2

The Godunov scheme for the classical LWR model on the line

The entropy solution of (1) is the only solution considered usually in the literature on traffic flow modelling. In entropy solutions, the decelaration of trafic generates Shockwaves, whereas the acceleration of trafic induces rarefaction waves. Entropy solutions are also characterized by the fundamental fact that they maximize locally the flow [Le 96]. The Godunov scheme [GR 91], [Kr 97] provides a numerical solution of the classical LWR model, as shown in [Le 96], [Da 95]. This solution is satisfactory for applications: it approximates the entropy solution. Let us introduce the equilibrium supply and demand functions:

(the symbols + and - represent right- and left-hand limits). The following illustration describes these functions, that represent respectively the greatest possible inflow (supply) and the greatest possible outflow (demand) at point x.

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With these notations, the expression of the Godunov scheme is straightforward:

with as usual (i) = [%i-\, Xi] the cell i of length li, t the index of the time-step, Kti the average density in cell i at time t At, Qti the average flow at point Xi during time-step t. The flow equation expresses that the flow is the minimum between the downstream supply and the upstream demand. The flow equation in (4) is the expression of the analytical solution of the Riemann problem, which can be obtained even if the upstream and downstream equilibrium relationships differ [Le 96]. The flow maximizing property of the entropy solution is crucial for this result. The practical necessity for considering space-dependent and even space-discontinuous equilibrium relationships is obvious: lane drops, intersections provide contexts for such discontinuities.

3 3.1

Extensions of the basic LWR model Link boundary conditions

For applications, extensions of this basic LWR model are indispensable. A first and obvious generalization concerns the extension of the model to networks, which implies two steps: the definition of proper boundary conditions for links, and the description of intersections. The equilibrium supply and demand concepts provide the proper framework for the definition of link boundary conditions. Considering now a link such as the following:

the boundary conditions are the upstream demand A u (£) and the downstream supply Sd(t) [Le 96], [LK 98]. The link inflow Q(a,t) at any time is the minimum between the link supply S e (K(a+, t ) , a ) and the upstream demand A u (t). Similarly, the link outflow Q(b, t) at any time is the minimum between the link demand Ae (K(b—,t),b) and the downstream supply £d(t). Thus:

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Partial flows on links

Before considering intersections, it is necessary to consider partial flows on links. Indeed, the fashion in which the traffic flow separates (according to preselection lanes) or does not separate (so-called FIFO flow) in the incoming links of an intersection determines the way in which the intersection works. Further, in many advanced applications, various categories of users must be considered: users would be distinguished according to destination, information availability, path, etc. The macroscopic variables are disaggregated according to some assignment attribute d:

The partial flows and densities are related by the trivial relationships:

These equations must be completed by a phenomenological model. The simplest possible model is the so-called FIFO model:

(vehicle speed independent of attribute d). This model results in a straightforward advection equation relative to the composition coefficients Xd — Kd/K of the flow: in which the global velocity of the flow V results from the resolution of (2)). A more realistic model is the lane assignment model [LK 98]. In this model, vehicles may have restricted access to lanes according to the assignment attribute d. Let / be the set of lanes, Id the set of lanes accessible to vehicles d, JiKmax the maximum density of lanes i, Kf the density of vehicles d in lanes i. Then the Kf are the unknowns of the lane-assignment problem and are subjet to the following constraints:

The Kd constraints express the split of Kd into the Kf, and the1/2Kmax constraints express that the total density in lane i cannot exceed the maximum density riK max of this lane. The Kd constitute the dynamic data and the

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7i and Id constitute the geometric data of the lane assignment problem. The unknowns K? can be determined by solving either

(maximizing locally the total flow), or

(Wardrop optimum), subject to constraints (9) in both instances. The meaning of (11) is to assign users to lanes in such a way that all users having the same attribute d have the same speed on all the lanes they use effectively (otherwise, users would switch lanes in order to drive faster: this is an individual optimum). These lane assignment models result in systems of conservation laws for which approximate Riemann solvers are under study. A simple case (2 user types, 2 lanes) was analyzed in [Da 97]. Supply-Demand models for partial flows can also be defined. The principle is to calculate partial supplies and demands for all superscripts d and to determine the corresponding partial flows by comparing partial demands to partial flows. Partial demands are defined as:

which is a FIFO-like model. The partial supply model really defines the user behavioral model. Let us first define coefficients fid which determine the maximum density (i.e. fidKmax) of the lanes available to vehicles d. If we refer to the notations of the preceding subsection,

We propose the following two models for the partial supplies: Model 1: £ d (z,f) = (3dY,(x,t) (linear model), Model 2: E d (z,£) = (3dZe ( K < 1 ( x + , t ] ,x) (homogeneous section model). Model 1 is extremely simple but allows Kd to exceed PdKmax. Model 2 does not have this drawback but still does not take partial flow overlapping into account as precisely as the lane-assignment models, since the data formed by the Id sets and the coefficients 7$ has been simplified and only coefficients (3d are left. It would also be natural to define the partial flows Qd as: Nevertheless, since ^j3d is usually > 1 (because of the partial flow overlapd

ping), it is possible that partial flows calculated according to (13) satisfy:

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thus implying a model inconsistency. This inconsistency can be resolved by using the following expressions:

The partial flows Qd can be viewed as solutions of the following program:

with (pd some concave increasing functions such as: I he above functions

3.3

Modelling intersections

Modelling intersections is more difficult, since intersection models are phenomenological by nature. They describe for instance, in the case of a merge, the local priority rules, or the gap acceptance process. Two modelling schemes can be considered. 1. Modelling intersection as objects of finite extension, by trying to reproduce the movement dynamics. This was the idea of the STRADA model [BLLM 96], in which exchange zones generalize cells inasmuch as they behave similarly but are endowed with several entry- and exit- points and provide upstream demands respectively downstream supplies for downstream respectively upstream cells or exchange zones. Exchange zone models are discrete by essence. 2. Pointwise intersection models. These were considered in [Le 96] [LK 98] and derived from zone models by letting the zone extension become vanishingly small. The study of these models is the subject of ongoing research. Let us still give one example and consider the node depicted hereafter. Let us denote Ej(t) the supply of exit link (j) of the intersection at the node point, and Aj(t) the demand of the entry link (i) of the intersection at the node point. A proportion 7ij of users about to exit link (i) chooses link (j) (the coefficients jij are called assignment coefficients and must be considered exogeneous to the flow model). Thus the partial demand of traffic from link (i) to link (j) def is given by: Ay-(t) = 7jjA;(£). Split supply coefficients bij (depending on the link geometry) can be introduced in order to disaggregate the link supplies £j(£). We can deduce partial supplies Sij(i) by applying for instance model def I (the simplest): %ij(t) — bijEj(t). Since it is possible and even likely that

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1, a formula similar to (14) should apply, with the same rationale, to yield the partial flows

4

Discretization of partial flows in links and intersections

Let us consider the Supply-Demand model for partial flows (notably equations (12) and (14). If we consider two consecutive cells (i}, (i + 1), the following relationships result (discretized model): (expressing the traffic supply of cell (i + 1) as a function of the cell mean density K\+l, (expressing the partial supplies, according to partial supply models 1 or 2, (expressing the traffic demand of cell (i) as a function of the cell mean density expressing the partial demands according to the FIFO-like model, expressing the partial flows between cells (i) and (i + 1) according to (14), expressing the discretized conservation equation, yielding the total cell density and flow as the sums of partial densities and flows. The similitude between the intersection model (16) and the Supply-Demand partial flow model (12), (14) is evident, thus the discretization of the intersection model follows the same lines as the discretized Supply-Demand partial flow model and need not be described in detail here.

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5

Conclusion

The development of extensions of the basic LWR is still an ongoing process. The only intersection model for which there exists any kind of experimental support is the Supply-Demand model [LK 98], and link partial flow models are still tentative. Nevertheless, suitable discretized models should be developed, in order to be able to choose between alternative modelling schemes. The finite volume method, combined with the search for analytical solutions, seems to be the best approach to the investigation of numerical solutions of the LWR model and its extensions.

References [BLLM 96] C. Buisson, J.P. Lebacque, J.B. Lesort, H. Mongeot. The STRADA model for dynamic assignment. Proc. of the 1996 ITS Conference. Orlando, USA. [Da 95] C.F. Daganzo. A finite difference aproximation of the kinematic wave model. Transportation Research 29B. 261-276. 1995. [Da 97] C.F. Daganzo. A continuum theory of traffic dynamics for freeways with special lanes. Transportation Research 31 B. 83-102. 1997. [GR 91] E. Godlewski, P.A. Raviart. Hyperbolic systems of conservation laws. SMAI. Ellipses (Paris). 1991. [Kr 97] D. Kroner. Numerical schemes for conservation laws. Wiley Teubner. 1997. [He 97] D. Helbing. Verkehrsdynamik. Springer Verlag. 1997. [Le 95] J.P. Lebacque. L'echelle des modeles de trafic: du microscopique au macroscopique. Annales des Fonts. 1st trim., 74: 48-68. 1995. [Le 96] J.P. Lebacque. The Godunov scheme and what it means for first order traffic flow models. Proc. of the 1996 ISTTT (J.B. Lesort ed.). 647-677. 1996. [LK 98] J.P. Lebacque, M.M. Khoshyaran. First order macroscopic traffic flow models for networks in the context of dynamic assignment. EURO Work Group on Transportation 1998, Goteborg (Sweden). CERMICS Report. To be Published. [LL 99] J.P. Lebacque, J.B. Lesort. Macroscopic traffic flow models : a question of order. 14th ISTTT. Accepted for publication. 1999. [LW 55] M.H. Lighthill, G.B. Whitham. On kinematic waves II: A theory of traffic flow on long crowded roads. Proc. Royal Soc. (Lond.) A 229: 317-345. 1955. [Ph 79] W. F. Phillips. A kinetic model for traffic flow with continuum implications. Transportation Planning and Technology, Vol. 5, 3, pp 131-138. 1979. [PH 71] I. Prigogine and R. Herman. Kinetic theory of vehicular traffic. American Elsevier, New York. 1971. [Ri 56] P.I. Richards. Shock-waves on the highway. Op. Res. 4: 42-51. 1956. [Sc 88] S. Schochet. The instant response limit in Witham's non linear traffic model: uniform well-posedness and global existence. Asymptotic Analysis. 1, pp 263-282. 1988.

Finite volume method applied to a solid/liquid phase change problem

El Ganaoui M., Bontoux P. IRPHE-Umversite d'Aix-Marseille II IMT, 38 Joliot Curie 13451, Marseille Mazhorova O. Keldysh institute Moscow

ABSTRACT A second order accuracy method of time and space based on finite volume approximation in a fixed m.esh is developped for Navier-Stokes and energy equations extended to solid/liquid phase change problems. This fixed grid method validated with respect to an interface tracking method is able to describe the interaction of steady and oscillatory melts with the interface during Bridgman crystal growth. Key Words: Finite volume, phase change, interface.

1. Introduction Free and moving boundary problem requires the simultaneous solution of unknown field variable and the boundaries of the domains on which these variables are defined. Phase change during directional solidification of semiconductor crystals by the Bridgman technique is a typical example of such a complex process. Each method of solution must solve the appropriate heat, mass and momentum transfer equations and determines the melt solid inter-

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face. It was still necessary to satisfy the Stefan or similar derivative condition on that boundary. Furthermore it was sometimes be difficult or even impossible to track the moving boundary directly [CRA 84]. The possibility, therefore, of reformulating the problem in such a way that the transmission conditions at the interface are implicitly bound up in a new form of the equation, which applies over the whole of a fixed domain. The moving boundary appears, a posteriori as one feature of the solution. One possibility of reformulating the problem is to introduce an enthalpy function in the energy equation and a porous model in the momentum equation. The enthalpy function is the sum of the specific heat and the latent heat required for the phase change. In the momentum equation, we assume that the liquid turn to solid in an intermediate region to be a porous medium. In this way on prescribing a Darcy source term the velocity value arising from the solution of the momentum equation are inhibited, reaching values close to zero on complete solid formation [VOL 80]. The coupled enthalpy porosity model gives a single set of homogenous NavierStokes and energy equations adapted to the problem of phase change during directional solidification [MOR 99]. The finite volume method is validated with respect to an interface tracking method [ELG 96]. It uses a fixed grid and the interface position is given from the thermal field (solidification isotherm). The resulting interface shape is also studied and some insight on cristal constitution are given. 2. Formulation

For directional solidification, A cylindrical ampoule with radius R and length L contains melt and crystal. The ampoule must be moved relative to a prescribed external temperature gradient. This motion of the ampoule is acounted for by supplying a melt to the top of the computational space at a uniform velocity Ut and with drawing cristal from the bottom with the same velocity. The heat transport between the furnace and the ampoule is modelled with a prescribed furnace temperature profile with three zones, cold (T = T c ) , adiabatic (dT/dn = 0) and hot one (T = T/j). The length L, the velocity a/L and the thermal difference Th — Tc are used as reference scales to give dimensionless form of the variables ( x , u , 6 ) . x ( r , z) represents the courant point with radial r and the axial z components, u = (ur,uz] represents velocity with radial and axial coordinates ur and uz, respectively. Liquid, solid and intermediate medium are distinguished by the suffixes /, s and si. For the energy equation a continuous enthalpy function is introduced :

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where £ is a prescribed small regularisation of the temperature, /; a mesure of liquid fraction and Ste = Lj/c(Th — Tc) is the Stefan number. The corresponding enthalpy is continuous and piecewise linear. In this way the energy equation takes the following form in all the dimensionless domain 0 < r < 1 et Q

If Ste 1 goes to 0, the equation (2) goes to the classical one phase energy equation. For the momentum equations a permeability term is introduced:

The momentum equations takes the following form :

where Ra = gfl(Th — T c )/i/a, is the Rayleigh number, Pr — i//a is the prandtl number characteristic of the material and Da = L 2 /A'o is the Darcy number characteristic of the phase change morphology. This equation contains asymptotically the behaviour of the velocity in each media. When A' = A"/, the term K(x,t) lu tends to zero and in the fluid domain QI and we solve a near approximation of the Navier Stokes-equations. When K = K s , the term K(x,t)~1u penalises the momentum equation, the other terms of the equation become negligible and implie us ~ 0. Equations (2) and (4) must be added to the continuity equation:

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3. Solution method

3.1 Finite volume method Equations (2), (4) and (5) are discretized by the finite volume method [EYM 97]. To explain the approach we consider the two-dimensional convection diffusion equation for a general variable (p with the velocity field in cartesian coordinates u :

f ( ( f ) = u/f — 7^3 grad