Lecture Notes in Computational Science and Engineering Editors T. J. Barth, Moffett Field, CA M. Griebel, Bonn D. E. Keyes, Norfolk R. M. Nieminen, Espoo D. Roose, Leuven T. Schlick, New York
27
Springer Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo
Siegfried Mtiller
Adaptive Multiscale Schemes for Conservation laws With 58 Figures
Springer
Siegfried Muller Institut fur Geometrie und Praktische Mathematik RWTHAachen Templergraben 55 52056 Aachen, Germany e-mail:
[email protected] Catalog ing- in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; det ailed bibliographic data is available in the Internet at .
Mathematics Subject Classification (2000): 65M12, 65M55,42C15, 47A20, 76Axx, 35L65 ISSN 1439-7358 ISBN 3-540-44325-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyr ight. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks . Duplication of this publication or parts th ereof is permitted only under th e provisions of th e German Copyright Law of Septemb er 9, 1965, in its current version, and permission for use must always be obta ined from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, regist ered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and th erefore free for general use. Cover Design: Friedhelm Steinen-Broo, Estud io Calarnar, Spain Cover production: design & production Typeset by the author using a Springer TEX macro package Printed on acid-free pap er
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46/3142/LK - 543210
To my par ent s
Preface
During the last decade enormous progress has been achieved in the field of computational fluid dynamics. This became possible by the development of robust and high-order accurate numerical algorithms as well as the construction of enhanced computer hardware, e.g., parallel and vector architectures, workstation clusters. All these improvements allow t he numerical simulation of real world problems arising for instance in automotive and aviation industry. Nowadays numerical simulations may be considered as an indispensable tool in the design of engineering devices complementing or avoiding expensive experiments. In order to obtain qualitatively as well as quantitatively reliable results the complexity of the applications continuously increases due to the demand of resolving more details of the real world configuration as well as taking better physical models into account, e.g., turbulence, real gas or aeroelasticity. Although the speed and memory of computer hardware are currently doubled approximately every 18 months according to Moore's law, this will not be sufficient to cope with the increasing complexity required by uniform discretizations. The future task will be to optimize the utilization of the available resources. Therefore new numerical algorithms have to be developed with a computational complexity that can be termed nearly optimal in the sense that storage and computational expense remain proportional to the "inherent complexity" (a term that will be made clearer later) problem. This leads to adaptive concepts which corr espond in a natural way to unstructured grids. The conclusion is justified by results of approximation theory which clearly indicate that nonlinear approximations, e.g., the positions of the discretization points are not a priori fixed, are more efficient than linear approximations, e.g ., uniform discretizations. For details on nonlinear approximation theory see [DeV98]. Currently, num erous effort s of this type are made in different research fields such as image processing, data compression, partial differential equations . In this monograph, the adaptation concepts for partial differential equations are of special interest which shall be bri efly reviewed . A naive technique is the remeshing of the grid where a fixed number of mesh points is relocated. Obviously this concept is aiming at balancing the error with a fixed number of points rather than reducing the error to a given tolerance. In order to meet a fixed error tolerance the grid adaptation has to allow for
VIII
P reface
m esh enrichme n t, i.e., locally refining and coarsening the mesh. This may result in an un structured grid with locally han ging nodes. Inst ead of refining t he grid it is also possible t o increase locally t he approxima t ion ord er p or apply a different discreti zation operator for a fixed grid. This leads to a hybrid discretization. Of course both st ra teg ies can be combined. More details on this subject can be found for inst an ce in [Sch98]. For ti me- dependent pr oblems one might also apply local tim e st eps. In t his case, the const raint for th e time discreti zation due to a CFL number is locally weakened without causing inst abilities. Hence the solut ion may evolve fast er in t ime for coa rse cells t han for fine cells. Of course , the solut ion has to be synchronized in case of inst ationary pr oblems but not necessaril y for steady state problems. For det ails see e.g. [B084] . Inst ead of ada pting the discretization one might also locally change the underlying m odel, e.g. lineari ze the mod el or neglect higher order derivatives if t he corres ponding physical effects are small. Alt hough t he above t echniques differ in the ada ptatio n st ra teg y t hey have one pro blem in common, nam ely, the cont rol of th e adaptat ion. Two st rategies t hat are applied in the conte xt of grid refinement shall be bri efly summa rized. Here we distin guish between concepts based on error indicators and error estim ators, respectiv ely. In case of err or indi cators, t he grid is remeshed , e.g., according t o stee p (discret ely approximated) gra dients of a physically relevant quanti ty or ot her indic ators. However , t his st rateg y provides only cont rol on t he grid refining and coarsening but no inform ation about the err or of t he approximation. A reliable concept is the error- balancing strategy. The goal is to equilibra te th e error. To t his end, a toleran ce tal and a maxim al number of discreti zation points N m a x are fixed . By means of residu al-based a post eriori est imates th e grid is locally refined until a local error est imator is prop orti onal t o the ratio tal/Nm a x . This lead s t o an opt ima l mesh size distribution. In pr act ice, it cannot be realized. Therefore one is aiming at an almost qu asi equidist ribution of th e error t olera nces. Numerou s results on a posteriori error est imates have been reported in the lit erature for elliptic problems, see [Ver95, EEHJ95, BR96 , HR02], par aboli c problems [EJ91 , EJ95] and hyp erbolic problems see [Tad 91, CCL94 , Vil94, JS9 5, CG96 , Noe96, SH97, K099]. During the last decad e new st ra teg ies have been developed based on multiscale techniques. Here wavelet techniques have become very popular. The basic idea is t o decompose t he t rial space into a coarser approximation space and a complement space spanned by so- called wavelet functions. This decomposition is recursively applied t o the coar se approxima t ion space. Finally, we obtain a decomposition of t he t rial space into t he coarsest app roximat ion space an d a sequ ence of complement spaces represent ing the difference between t he approxima t ion spaces . Performing a change of basis t he solut ion can now be equivalent ly repr esent ed in te rms of t he single- scale basis corres ponding to t he t rial spac e of t he finest approximation space and the multi scale or wavelet basis, respectively. Since the coefficient s of t he wavelet expansion , so-called wavelet coefficient s or det ails, may become sma ll whenever the so-
Preface
IX
lution is locally smooth, data compression can be performed applying threshold techniques. For instance, one only keeps the N largest coefficients . Here the objective is to minimize the error by N coefficients (see e.g. [CDDOl]). This corresponds to the idea of best N -term approximation. Alternatively, a tolerance e can be fixed and all details smaller than this threshold value are discarded. Here the idea is to reduce the total number of coefficients to a small number of significant coefficients where the error to the approximate solution of the underlying approximation space is proportional to e (see e.g. [GM99a , CKMPOl]). In order to control the threshold error we need to relate coefficient norms to function norms. The present work is concerned with developing and analyzing an adaptive finite volume scheme (FVS) for the approximation of multidimensional hyperbolic conservation laws. The concept is based on multiscale techniques which have already been mentioned above. First work on this subject has been reported by Harten [Har94, Har95] . Here the goal is the acceleration of a given FVS on a grid of uniform resolution by a hybrid flux computation. The core ingredient is the multiscale decomposition of a sequence of averages corresponding to a grid of finest resolution into a sequence of details and coarse grid averages. This decomposition is performed on a sequence of nested grids with decreasing resolution. It can be utilized in order to distinguish smooth regions of the flow field from regions with locally strong variations in the solution. In particular, the hybrid flux evaluation can be controlled by the decomposition, i.e., expensive upwind discretizations based on Riemann solvers are only applied near discontinuities of the solution. Elsewhere cheaper linear combinations of already computed numerical fluxes on coarser scales are used instead. These correspond to finite difference approximations. In the meantime this originally one-dimensional concept has been extended to multidimensional problems on Cartesian grids [BH97, CDOl], curvilinear patches [DGMOO] and triangulations [SSFOO, Abg97, CDKPOO] . The bottleneck of Harten's strategy is the fact that the computational complexity, Le., the number of floating point operations as well as the memory requirements, corresponds to the globally finest grid. In view of multidimensional applications, this is a severe disadvantage. Recently, a real adaptive approach has been presented in [GM99a] and has been investigated in [CKMPOl] where the computational complexity is proportional to the problem-inherent degrees of freedom. The basic idea of this concept is to determine an adaptive grid by means of a sequence of truncated details . The set of significant details can be interpreted as a tree. Then the adaptive grid is constructed by locally refining the grid according to the tree of significant details. This leads to an unstructured grid with hanging nodes. In order to restrict the computational complexity to the number of significant details the multiscale transformation is only performed on the set of significant details and the averages corresponding to the adaptive grid. It turned out that the grading of the tree simplifies the local transformation without increasing the
X
Preface
complexity. In particular, the leaves of the graded tree directly correspond to the adaptive grid. In order to preserve the accuracy of the reference FVS with respect to the finest grid the numerical fluxes on the adaptive grid have to be evaluated judiciously. No error at all is introduced when locally performing the flux evaluation by means of the averages on the finest scale. However, this requires a local reconstruction process by which the computational complexity is increased for multidimensional problems. Investigations for a one-dimensional scalar equation verify that for first order approximations the accuracy of the adaptive FVS is much less than that of the reference FVS (see [CKMPOl]) . However, parameter studies show that in case of higher order accurate FVS based on reconstruction techniques this constraint can be weakened. Here it is possible to utilize the given local averages directly instead of computing the averages on the finest scale . The target accuracy is still preserved by means of the solver-inherent reconstruction step. A point of special interest is the reliability of the scheme, i.e., the perturbation error introduced by the truncation process can be controlled over all time levels. For this purpose analytically rigorous estimates have to be derived by which the details on the new time level can be estimated by those already computed in the previous time step. For the one-dimensional scalar case this prediction has been analytically investigated in [CKMPOl]. The results derived there justify for the first time the heuristic approach suggested by Harten. By now the new adaptive multiresolution concept has been applied by several groups with great success to different applications, e.g. , 2D-steady state computations of compressible fluid flow around air wings modeled by the Euler and Navier-Stokes equations, respectively, on block-structured curvilinear grid patches [BGMH+Ol], non-stationary shock-bubble interactions on 2D Cartesian grids for Euler equations [MiiI02], backward-facing step on 2D triangulations [CKP02] and simulation of a flame ball modeled by reaction-diffusion equations on 3D Cartesian grids [RS02]. This book presents a self-contained account of the above adaptive concept for conservation laws. The main objectives are the construction and the analysis of the local multiscale transformation, the derivation of the adaptive FVS and a rigorous error analysis. New applications on Cartesian and curvilinear grids for the 2D Euler equations are presented which verify that the solver can be applied to real world problems. According to this the outline of the present work is as follows: In Chap. 1 the governing equations ar e presented and some of the characteristic properties are summarized. This is concluded by a brief introduction to Godunov-type schemes which form an important class of FVS frequently applied to approximate the solution of conservation laws . The multiscale setting is outlined in Chap. 2. It is based on a hierarchy of nested grids . As a simple but important example the Haar basis is pr esented to outline the basic principles and the goal of the multiscale
P reface
XI
set ting. This motivat es th e genera l framework of biorthogonal wavelets and st able completi ons. Modifyin g th e Haar basis appropriate ly lead s to a new basis wit h "go od" cancellation prop erties which is utilized in the adapti ve scheme. In Chap . 3 th e local multi scale analysis is introduced by means of t he modified basis. In particular, the tree of significant details , the gra ding of t he t ree and t he const ruction of t he ada ptive grid are investigated in some det ail. The perform an ce of th e local multiscale transformation is an alyzed in det ail which results in sufficient condit ions for the gra ding of th e details. The construction of t he adaptive FVS is presented in Chap. 4. In particular , several strategies for t he evaluation of the numerical fluxes are discussed and t he construction of t he prediction set of significant det ails on the new time level is outlined. An err or ana lysis is present ed in Chap . 5. It is based on an ansatz origina lly considered by Har ten [Har95] in the context of his hybrid scheme and t he result s derived in [CKMP01] . An efficient implement ation of t he ada ptive scheme crucially depends on the data st ruct ures by which the algorit hm is realized. This is no longer a t rivial t ask as it is for schemes based on st ructure d meshes. In order to realize optimal computationa l complexity the dat a st ructure s have to be ada pte d judiciously to t he und erlying ada ptive algorit hm. Such appropriate dat a structures are discussed in Chap . 6. Finally, in Chap . 7, some relevant num erical exa mples illust rate t he computationa l complexity and accuracy behavior of t he scheme and problems arising in engineering applications are present ed . Acknowledgment s: It is a great pleasur e for me to express my gra tit ude to those persons who have been supporting my scientific work . In particular , I wish to thank my three mentors: first of all, Prof. Wolfgang Dahmen , RWTH Aachen , who introduced to me the world of wavelets and showed me th e mathematical concepts beyond th e technical det ails; furthermore, Prof. Jos ef Ballm ann , RWTH Aachen, who depicted to a math ematici an the physics behind the mathemati cal mod els and last but not least Prof. Rolf J eltsch, ETH Zurich , for his ent husiasm and optimism encouraging me to start with a scientific car eer. Moreover , I would like to th ank my colleagues at t he Institut fiir Geometrie und Prakti sche Mathematik, RWTH Aachen. Among ot hers I would like to point out Dr . K.-H. Brakh age for his neverending help concern ing any kind of software relat ed problems, Dipl.-M ath. Alexander VoJ3 for discussions on software concept s and t he design of data st ruct ures and Frank Knoben for his invaluable work as syst em administrator. The present work was supp orted in parts by th e collabora tive resear ch center SFB 401 "Modulation of Flow and Fluid-Structure Interaction at Airpl ane Wings" and t he EU-TMR Network "Multiscale Methods in Numerical Simulati on" . The latter made it possible to spend six months at the Lab oratoire d'Analyse Numer ique, Universite Pierr e et Mari e Curi e, Paris VI. Thi s resear ch st ay had a st rong influence on my scientific work. In particular , I t ha nk my collaborators Prof. Albert Cohen, Dr. Sidi M. Kab er and Dr. Marie Post el. Furtherm ore, I would like to thank Dipl.-Ing. Frank Bramkamp and Dipl.-
XII
Preface
Math. Philipp Lamby for their cooperation in developing the new flow solver QVADFLOW which verifies that the present adaptive multiscale concept is a useful tool in solving efficiently and reliably real world problems. Prof. Wolfgang Dahmen and Prof. Sebastian Noelle, RWTH Aachen, and Prof. Thomas Sonar, TV Braunschweig, acted as referees for my habilitation thesis , and I would like to thank them for their careful reading of my work. The constructive comments of several unknown referees contributed significantly to the final version of the present book. I would also like to thank Dr. Martin Peters and Thanh-Ha Le Thi, Springer Verlag, for the professional and pleasant . cooperation. Last but not least I would like to express my deepest gratitude to my wife and colleague Dr. Birgit Gottschlich-Miiller for her collaboration and her love. She always encouraged me to continue with my work.
Aachen, September 2002
Siegfried Miiller
Table of Contents
1
Model Problem and Its Discretization . . . . . . . . . . . . . . . . . . . . 1.1 Conservat ion Laws 1.2 Finite Volum e Methods
1 1 6
2
Multiscale Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.1 Hierar chy of Meshes " 2.2 Motivation... ........ ... . ....... ..... .. ... . . .. . ....... 2.3 Box Wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.3.1 Box Wavelet on a Cartesian Grid Hierar chy. . . . . . . . .. 2.3.2 Box Wavelet on an Arbi trary Nested Grid Hierar chy . . 2.4 Change of St able Completion ... 2.5 Box Wavelet with Higher Vanishing Moment s . . . . . . . . . . . . .. 2.5.1 Definition and Construction 2.5.2 A Univariate Exampl e ....... 2.5.3 A Remark on Compression Rates . . . . . . . . . . . . . . . . . . . 2.6 Multiscale Tr ansformati on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 11 13 17 17 19 22 24 24 26 29 29
3
Locally Refined Spaces 3.1 Adaptive Grid and Significant Details . . . . . . . . . . . . . . . . . . . .. 3.2 Grading .. . . . . . . . . . .. . . . . . . . ... . . . . . .. . .. .. . .. . . .. . . . . . 3.3 Local Multiscale Transform ation 3.4 Grading P ar ameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Locally Uniform Grid s ............................ 3.6 Algorithms: Encoding, Thresholding, Gradin g, Decodin g . . . . 3.7 Conservation Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Appli cati on to Curvilinear Grid s
33 34 36 44 47 52 55 60 62
4
Adaptive Finite Volume Scheme 4.1 Construction . . . . .. .. . . . . . .. . .. . .. .... . . . ... . ... ....... . 4.1.1 Strat egies for Local Flux Evaluation 4.1.2 Strat egies for Prediction of Details 4.2 Algorithms: Initial dat a, Prediction, Fluxes and Evolution ..
73 73 75 77 82
Table of Contents
XIV
5
Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Perturbation Error 5.2 Stability of Approximation 5.3 Reliability of Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
89 90 93 97
6
Data Structures and Mem or y Managem e nt 6.1 Algorithmic Requirements and Design Criteria 6.2 Hashing 6.3 Dat a St ru ct ures
113 113 115 118
7
Numerical Experiments 7.1 Parameter St udies 7.1.1 Test Configurations 7.1.2 Discretizat ion 7.1.3 Compu tati onal Complexity and St ability 7.1.4 Hash P arameters 7.2 Real World App lication 7.2.1 Configurations 7.2.2 Discret ization 7.2.3 Discussion of Resu lts
123 123 124 126 127 131 133 133 134 136
A
Plots of Numerica l E xperiments
139
B
The Context of B io r t hogon al Wavelets B.1 General Setting B.1.1 Multiscale Basis B.1.2 Stable Comp letion B.1.3 Multiscale Transformation B.2 Biort hogonal Wavelets of the Box FUnction B.2.1 Haar Wavelets B.2.2 Biort hogonal Wavelets on the Real Line
151 151 152 153 154 157 157 158
R eferences
163
List of Figures
169
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Index
179
1 Model Problem and Its Discretization
The objective of this chapter is the introduction of conservation laws and how to approximate their solution. Basic aspects concerning existence and uniqueness are briefly reviewed . Motivated by analytical considerations finite volume schemes based on Godunov-type methods are summarized which ar e frequ ently applied in numerical simulations.
1.1 Conservation Laws The present thesis is concerned with evolution equations of the form ou(t,x) ot
d
' " ofi(u(t,x)) _ +~ ox - 0, i=l
(t ,x) E (O,T) x
n
(1.1)
'
describing the temporal change of the conservative quantities u : [0, T] x n -+ V due to spatial variations quantified by the fluxes f i : V -+ Rm, i = 1, . . . , d, for each of the spatial directions. Here the open set neRd denotes the computational domain and V C R'" the space of admissible states. In the literature , evolution equations of the type (1.1) are referred to as scalar (m = 1) or system (m > 1) of conservation laws. They arise from the principles of balancing mass, momentum and energy in classical continuum mechanics under local regularity assumptions. In order to admit physically meaningful discontinuities, it is mor e convenient to consider the evolution of an averaged conservative quantity. By the divergence theorem , the partial differential equation (1.1) can be transformed into the ordinary differential equation d dt
r
lv
u(t ,x)dx=-
r
lev
fn(x)(u(t,x))dx ,
(1.2)
where V is any (time-independent) control volume with Lipschitz continuous boundary and n denotes the outer normal to OV. The flux in normal direction is defined by d
fn(u) :=
L fi(u) ni · i=l
2
1 Model Problem and Its Discreti zation
Obviously, the qu antity Ev(t) := Iv u(t , x) dx remains const an t if the right hand side in (1.2) vani shes, i.e., the incomin g and outgoing information across the boundar y aV are balan ced. Not e, that the evolution equat ion (1.2) corresponds to t he bal anc e equa t ions in cont inuum mechanic s. Later , this equation will be t he start ing point for the design of an appropriate numerical scheme. Throughout t his work t he fluxes are assumed to be smoot h, i.e., f i E C 2 (D, R'") and t he J acobian of the normal flux d
._ afn(u) _ """" afi(u) . A(u ,n ) .au - ~ au n.
(1.3)
i= l
has m real eigenvalues Ai(U, n) as well as m linearly ind epend ent righ t eigenvectors r i(u, n) and left eigenvect ors lieu , n) , i = 1, . .. , m , resp ectiv ely, for all u E D and nERd with IInl12 = 1. In t his case the syst em is called hyp erbolic. Some exa mples shall be pr esented in the following. Example 1. (Traffic Flow (m = d = 1), [LeV92 , Kro97]) Here u denot es t he t raffic density, i.e., cars per uni t length, and f eu) t he t raffic flow, i.e., cars per unit time. A simple model for t he flow is given by
feu ) = -(u - u m )2 + fo where the flow increases until a maximal speed U m, e.g. speed limit , is reached and t hen decreases. Example 2. (Bu ckley-Leveret t equation (m = d = 1), [LeV92 , Kr o97]) This is a simple model for two ph ase fluid flow, e.g. wat er and oil, in porous media such as sa nd, which arises in oil recovery when water is pumped int o an oil field forcin g out the oil. Here u is t he wat er saturation and the relative permeabili ty is given by
u2 fe u) = u2 + a(l _ u2)' wher e a is t he ratio of t he viscosity coefficients corresponding t o t he two ph ases. Example 3. (Eul er equat ions (m = d + 2, d 2: 1), [CF48 , CM79]) T hese equations are derived in gas dynam ics t aking only invis cid effects into account. The motion of the fluid is describ ed by the cont inuity equation, the momentum equation and t he energy equa t ion for conservation of mass, mom entum and energy. This results in a syste m of conservation laws . Her e u = (0, ill , aE) is t he vector of mass density 0, momentum ill E Rd and t he total energy per uni t volume aE. The corres ponding fluxes ca n be writte n in t he form
1.1 Conservation Laws
3
where e i E Rd denot es t he it h uni t vecto r in Rd. Obviously, t he resulting system has to be comp lete d by an add it ional equation for t he pr essur e p , t he so-called equation of state p = p(g, m , g E ). Two frequentl y used models are polytropic gas : p = ({ -1) (g E - 0.5 m 2 /g) , 'Y> 1 isentropic gas:
p = "'gI,
'"
>
°
To ens ure a uni qu e solution initi al condit ions and in case of a bounded dom ain n boundar y condit ions have to be imp osed , i.e.,
n,
u (O , x ) = u o(x) ,
x E
u (t , x ) = ur (t , x ),
x E r (t) Can , t E (0, T).
(1.4) (1.5)
In t he sequel the initi al value problem (1.1) , (1.4) is referred t o as IVP and t he init ial boundary valu e problem (1.1) , (1.4) , (1.5) as IBVP. Of course, the condit ions (1.4) and (1.5) have t o be chosen judiciously in agreement with existe nce and uniqueness result s for the IVP and t he IBVP, resp ectively. The number of boundar y condit ions J.l depends on t he at tached flow field . In par ti cular , at most J.l condit ions are admissible if t here exist J.l negative eigenvalues of A(u, n ) where u = limh--+O+ u (t , x - h n ) denot es t he limit of t he solution u at t he boundar y point x E an and n the corres ponding oute r uni t normal. Note, that boundar y condit ions are only admissible at inflow boundar ies (J.l > 0) bu t not at outflow boundaries (J.l = 0) . For more det ails on init ial and bo undary condit ions t he read er is referred to [Kre70, KL89]. In the following t he question of existe nce and uniqueness is addressed in order to emphas ize t he pro blems arising lat er when (1.1) is discreti zed. A first result shows t hat a differentiabl e or classical solution in general only exists for small t ime even if the initi al dat a are smooth. Theorem 1. (Local Existence of classical solution) Let m = 1, d 2: 1 and n = Rd. Let Ii E ( 2(R, R), i = 1, . .. ,d, and Uo E C 1 (Rd , R) be bounded such that
{f ())auo( . . a x) } > Then there exists °such that a classical solution of the IVP exists for all 11 . f J.li,j:= m
ll (
x ERd
i
Uo x
Xj
- 00 , Z, J
= 1, .. . , d .
t E [0, 0,
(1.13)
The solut ion of t his pr oblem is known to be self-similar, i.e., it is constant along rays x / t = canst, and can be repr esented as u (t , x) = U (UL , UR , x / t) . For det ails on t he existence and t he const ruction of t he Riemann solut ion see [GeI59] (scalar) and [Liu75] (system) . Th en th e Godunov flux is det ermined by U(Vk , VI , 0) where t he initi al valu es ar e th e averages on both sides of the interface n,l ' Obviously, t his leads to a first ord er approxima t ion
since t he exact averages U k and UI are only first order approximations of t he point valu e u (t , x~ ,I ). If t he solut ion u would be known, t hen no erro r is int roduced when evaluating t he num erical fluxes at V = w = u (t ,x~ ,I) and n = n k ,1 (x~ ,I) according to t he consiste ncy of t he flux approximation. Solving such local Riemann problems accounts for a substant ial par t of t he overa ll computational effort . A less expensive approximation is base d on lineari zing t he Riemann pr oblem in the sense t hat t he flux f n is replaced by
1.2 Finite Volume Methods
9
the linear approximation fn(u) := A(u, n ) u with respect t o an int erm ediat e state u . This was first introduced by Roe [Roe81]. For mor e det ails on approximat e Riemann solvers see [Tor97]. Step 3: R econstruction. The low accuracy of the Godunov and Roe flux approximation can be remedied by means of reconst ruction techniques int ro duced by Hart en et . al [HEOC87] . The basic idea is to construct an approximation t o a scalar function w : [l -+ R based on its averages {wkh such t hat the corres ponding reconstructi on fu nction R N R (' , w) has a desired discretization error, satisfies t he conservation property and is essenti ally nonoscillat ory (E NO)
- R NR(X,W) = w(x ) + O(h NR),
w sufficient ly smooth,
- IJkl JVkR NR(X, w) dx = Wk, - TV(R NR(., w)) :S TV(w) + O(h C» ,
a> 1.
It should be emphas ized th at t he reconst ruc t ion funct ion is not necessarily
smoot h on the whole computat ional domain [l but may jump, for inst an ce, at t he int erfaces of t he cells Vk . In pr act ice, this does in fact happ en, since for each cell a local reconstruction function R ( ' , w) is det ermined by mean s of the averages in t he local neighbo rhood . In view of th e ENO property t he reconstruction ste ncil has to be chosen judiciously employing dat a depend ent possibly un symmet ric stencils. For the ID case an efficient strategy was developed in [HEOC87] by successively increasing t he reconstructio n st encil. For t he multidimensional case this turns out to be not feasibl e. Here a fixed stencil is in general chosen where th e reconstruction conditions are appropriately weight ed when solving a least- squ ares pr oblem (see e.g. [HC91, Abg91]). For more det ails on reconstruction t echniques, see also [Son95]. The reconstruction concept has originally been developed for scalar fun ct ions. It can easily be exte nded to syste ms by applying t he reconstruct ion component wise or vect orwise. More sophisticated techniques are based on t he reconstructi on oft he cha racteristic vari ables ui; := (li (U, n) , u) , i = 1, . . . , m , where li(U, n) denot es the left eigenvect or of t he J acobian A(u , n) . Now t he approxima t ion ord er of t he flux approximation can be increased . To t his end, t he averages are replaced by point valu es det ermined by th e loca l reconstruction funct ions Rr:R(. , u(t , .)), i.e.,
Rr:
G(Rr:R(x7,l, u(t, ')) , R{"R(x7,1, u (t , ')), n k,l) = f n k • , (x 7' 1) (u(t ,x7,l))
+ O(h NR).
Note , t hat for a first order reconstruct ion the point values coincide with the averages according t o t he conservat ion property of t he reconstruction. Step 4: Quadrature of time integral. Fin ally the time int egration has to be resolved. One possibility is t o apply a Runge-Kutt a scheme to t he evolutio n equation (1.2), incorp orating t he approximations due to th e precedin g
10
1 Model Probl em and It s Discreti zati on
steps. T his requires averages on some int erm ediate t ime levels which all have to be stored in t he comp utation of one t ime step. In order to avoid t his addit ional amount of storage one can apply a different st rategy. For t his purpose, the time int egration is approximated by means of a quadrature formul a , e.g. by a Gau ss quadrature rul e, where t he fluxes have to be approx imated at t he Gaussian points t n ::; tf < ... < t Na ::; tn+l . Since the solut ion u is not known at t ime level t~ , an approximation of u (t~ , x~ ,l ) is determined where u is expanded in a Taylor series at t he point (tn, xd . Here Xk denotes the cent roid of t he cell Vk . Then th e t ime derivatives are successively substit uted by spatial derivatives exploiting t he evolut ion equation (1.1). At t ime level t n t he averages are known. Hence t he spatial derivatives can be approximate d by t he reconstructi on function due t o t he known averages . Obviously, the computation of t he num erical fluxes is very expensive. Therefore one is int erest ed in reducing th e cost. Since upwind t echniques are only essent ial close to discontinuities, cheaper finit e difference approximations are sufficient in th e smooth part of th e solution. This requires a tool by which the data representing t he flow field can be analyzed. In t his regard Hart en introduced a sophist icated st rategy based on multiscale decompositio ns [Har9 5].
2 Multiscale Setting
The core ingredient of t he adaptive FV S is t he multiscale setting by which t he data at han d, here cell averages, can be analyzed . In t he lit erature, at least two settings are known which can be app lied for t his purpose. One framework was int roduced by A. Har t en [Har9 3a, Har93b, Har 96] and extended lat er by [ADH98, ADH99]. Here t he construction of a multi scale analysis is based on discret e dat a only and employs reconstruction and pr edicti on techniques. In view of stability invest igations, funct ion spaces have to be introdu ced in order to benefit from functi onal analytic arguments . Therefore we pr efer a different meth odology by W . Dahmen [Dah94 , Dah9 5, Dah96] and collaborators [CDP96] which is based on biorthogonal wavelet s and stable complet ions . Alt hough t he set t ings are different, t he resulting t ra nsformations can be rewrit ten in te rms of t he ot her setting, see [GM99b]. In the appendix of t his book , we pr ovide a short summary on wavelet t heory for readers who are not familiar with wavelets at all, see Sect. B. In t he following we present a self-contained multiscale setting employed in t he construc t ion of ada ptive multiscale FVS . In Section 2.1 we introdu ce a hierarchy of nest ed grids. By means of t he box funct ion and t he box wavelet we will motivat e t hat t he cell averages relative to t he nest ed grids are naturally relat ed to biorthogonal wavelets, see Section 2.2. We will explain in Sect ion 2.5 how to modi fy t he box wavelets such t hat t he new wavelets have imp roved cancellation pro pert ies. For th is purpose we pr esent a systematic fashion how to const ruc t the modified box wavelets. This is related to stable comp let ions, see Sect ion 2.4. Finally, we det ermine a mult iscale decomp ositio n of the cell averages relat ed t o a finest resolut ion level that will be utilized in Chapter 3 t o construct t he adaptive grid.
2.1 Hierarchy of Meshes As has been motiv at ed in th e pre vious cha pte r FVS are naturally relat ed to cell averages of t he solut ion. In ord er t o det ect singularities of the solut ion by mean s of t he array of averages, we consider t he difference of averages corresponding to different resolut ion levels. For t his purp ose we introdu ce a hierarchy of nest ed grids.
12
2 Multiscale Setting
DEEm j =O
j = l
j = 2
Fig. 2 .1. Sequ en ce of nest ed grids
Definition 3. (Nest ed grid hierarchy) A sequence of grids 9j := {Vj,k} kEfj ' j = 0, . . . , L , is called a nest ed grid hierar chy if the follow ing con diti ons hold
- n=
Uk Efj
Vj,k with
(partition)
IVj,k n Vj ,k,1 = 0 for k - Vj,k =
Ur EM j ,k Vj +l ,r ,
f:.
k' , k ,k' E I j , kE
t;
(refine me nt)
Note, that th e cells are always assumed to be closed.
Here t he coarsest grid is indicat ed by 0 and th e finest grid by L . Furthermore t he ind ex sets M j,k corre spond to t he new cells on level j + 1 resulting from t he refinement of t he cell Vj,k which is always assumed to be closed. A simple example is shown in Fig. 2.1 where a coarse grid is successively refined with increasing refinement level. From t he conditions of t he nested grid hierar chy we imm ediately conclude t hat t he following pr operties hold for t he refinement sets Mj ,k: - Mj ,k n M j,k' = -
U kEfj
M j,k =
0 for k f:. k' , k , k' E I j ,
(redunda ncy- free) (gap-free)
I j+ l.
For t he num erical experiments perform ed in Chap t er 7 only st ructured grids are considered. However , t he fram ework pr esent ed here can also be applied to unstructured grid s and irr egular grid refinements . For reasons of simplicity and stability only uniform refinements are considered here, i.e.,
# Mj ,k = M; = canst. Relative to the grids 9j we introdu ce the so- called box fun ction _
epj,k
()._ X
,-
1
V-i .kIIXv
" k
( ) _ { IVj,k\ - l ,
x -
0
,
x E Vj ,k X d V. l" J,k
(2.1)
defined as t he £I - scaled cha racteristic function with respect to Vj,k, i.e., Ilepj,kll£l(Q) = 1. Obviously, t he functions corr espond ing to t he same discret ization level are linearl y ind epend ent . The nest edn ess of the grids as well as t he lineari ty of int egration imply t he two-scale relation
2.2 Mot ivat ion
13
{;j ,k
~ Fig. 2.2 . Box function and box wavelet
r{Jj ,k
=
L r E M j ,k
IVJ+l,rl IV.J ,k I 'Pj+ l, r>
(2.2)
i.e., the coarse grid box function can be represent ed as a linear combination of the corresponding fine grid box functions. Consequ entl y, th e box functions can successively be computed st arting with the finest level. The reason for introducing the box functions is motivated by the fact that t he averages of a scalar, int egrabl e function u E £1(D) can be interpreted as an inn er product U j ,k
(2.3)
= (u, r{Jj,k ) n
with the box function where t he inner product is defined by
(u, v)n :=
l
u vdx.
Obviously, th e average s of two discretizat ion levels are related by U j ,k
=
L r E Mj ,k
IVJ+l,rl IV.J,k I U j +l ,r' A
(2.4)
The goal is to transform th ese dat a into a different form at of cell average s corres ponding t o a sequence of resolution levels. This will be motivated by a simple univari at e example.
2.2 Motivation We now consider the uni t interval D = [0,1] where th e grid hierar chy is determined by a uniform dyadi c partition of [0,1], i.e., VJ,k = 2- j[k , k+l]' k E I j := {O, . .. , 2j - I} . Not e, t hat the refinement sets are Mj ,k = {2k, 2k + I} . Then the £l - scaled box fun ction has the form
14
2 Mult iscale Setting {(,' k
'r
s,
= 2jX[0,1](2 j. -k) .
(2.5)
In the sequ el, we will explain how to decompos e th e cell averages into averages of a coa rser partition and details. To this end , we consid er in analogy to (2.2) the two-scale relation (2.6) by which a coarse- scale box function is reexpressed in t erms of fine-scale box functions . We now introduce the box wavelet -
'l/Jj ,k :=
1
'2 ( rpj+l ,2k -
rpj+l ,2k+d ·
(2.7)
Not e, th at the L 2 - scaled counterpart coincides with the Haar wavelet, see [Haa10] and Sect . B .2.1. Then we can write any fine-scale box function by means of the box function rpj ,k and th e box wavelet ;} j ,k . (2.8)
These relations are motivated by the illustrations in Fig. 2.2. In an alogy to (2.6) the cell averages satisfy A
Uj ,k
=
A A ) '12 (Uj+l ,2k + Uj+l ,2k+l .
On the other hand, (2.8) means
wher e the details are defined by dj ,k :=
(u, ;} j,k ) [O,l ] =
~ (Uj+l ,2k -
Uj+l ,2k+d ·
(2.9)
This relation shows how to reexpress fine-scale averages from coars e- sca le ones and det ails. In the sequ el, it will be convenient to rewrite the two-scale relations in matrix-vector form . To t his end , we introduce th e vectors (Pj := (rpj ,khEI; and .pj := ( ;}j,k hEI; ' Later we will use this notation also in the sense of a collect ion of functions . Then (2.6) and (2.7) read (2.10) where the columns of th e so-called mask matrices Mj,o and Mj,l contain the filter coefficients ~ , ~ and ~, - ~ of rpj ,k and ;}j,k, respec tively, i.e.,
In ana logy, t he two-scale relation (2.8) becomes (2.11) where the mask matrices are determined by Gj,i = 2 ML, i = 0,1. Note, that the relations (2.10) and (2.11) realize a change of basis , becaus e the composed matrices (2.12) are inverse, i.e.,
Gj =
Mjl . Hence, t he single blocks fulfill (2.13)
We now introduce a dual syst em by th e functions j j r i: , k := 2- ,ii r s,· k = X[0,1](2
in
.
-k) ,
'l/Jj ,k := T j ,(fj ,k
or in vector form Pj := 2- J Pj and tP'j := 2- j ~j. These are the L oonormalized counterparts of the box function and the box wavelet , resp ectiv ely. Obviously, the duals also satisfy two- scale relations of the form (2.10) and (2.11) with matrices M j ,i and Gj,i , i = 0,1. They are related to (2.12) by - T
M j ,i = Gj,i
From t his we infer that
,pj
and
- T
Gj ,i = Mj,i '
U ~j and Pj U IJtj ar e biorthogonal, i.e.,
(Pj , ,pj)[O,l] = (lJtj , ~j) [O,ll = I, (Pj , ~j) [O,l]) = (lJtj , ,pj) [O,l j = 0
(2.14)
where we use the notation (8 ,p):= ((B ,cp))OEl3 ,: e,r eEE,rEMj ,k'
# E)
are defined by (2.23)
In order to realize a change of basis, i.e., we obtain two-scale relations of t he form (2.8), we obviously need that th e matrix Aj ,k is invert ible, i.e. , t he inverse denoted by 1
Choos ing t he ansatz ;j;j,k,e
= 2::rEM j,k de',~ c,Oj+l,r
t he par amet ers de',~ are sub-
ject to t he conditions (1) cV..l1 , e E E*, i.e., {I, ;j;j,k,e )n = 0 and (2) {'ljJj ,k,e, ;j;j ,k,e/ )n = oe,e' Cj,k,e . These conditions hold for de',~ = a~',~ a{',~ .
20
2 Multiscale Setting - I . - (bj,k) Aj,k'r,e rEMj, k,eEE
exists. In this case we obtain (2.24)
However , we are not only int erest ed in performing a change of basis but th e resulting syst em of box functions and box wavelet s and their LCXl- scaled counte rparts defined by (2.25)
are biorthogonal. For t his purpose we prove th e following corollary. Corollary 1. (B iorthogonality of box wavelets) Assume th at th e vectors a~,k , e E E, form an orthonormal system, i.e., (a~,k) Ta~;k = r5 e ,e" Th en the systems ~j,k and'Itj,k .- ('l/Jj,k,e)eE E are biorthogonal, i.e., ('It j,k, ~j, k)!t = I. Proof. In ord er t o prove th e assert ion we consider first ( nl. 'f/j,k,e, .i'f/ j,k,e') !t
=
IVj,k I '" L..-
'" L..-
j,k ae,rao,r' j,k j,k a j,k () ao,r e , ,r' tpj+l,r , tpj+l ,r' o
rEMj ,k r'EMj ,k
for any k E I j , e, e' E E and j E No. By definition of the box function (2.1) we infer for r E M j,k and r' E M j,k l
_ _ ) s ( tpj+l,r, tpj+l ,r' o = Ur,r'
1
_, ( j,k)- 2 1 Ur,r' aO,r -IVI'
IV1 +I ,r I -
1,k
Note, that t he refinement sets are redundan cy- free. This impli es
for r -j. r' according to t he conditions of a nest ed grid hierar chy, see Definition 3. In addition, we use (2.21). Thus, we conclude by the assumpt ion th at nl' , nt, ) _ ( o/J,k,e, If/j ,k ,e' f2 -
'" ~
aj,k aj,k _ (a j,k )T a j,k _ s e ,T e' ,T e e , - U e ,e/
rEMj ,k
which proves th e assert ion. D ' k Note , that the vector a~' can be exte nded to an ort hogonal syst em by mean s of t he Gr am-Schmidt orthogonalization process. Fin ally, t he orthogonal vecto rs have to be normalized. According to t he univari at e case, we now int roduce t he det ails corre sponding t o th e box wavelet e E E*.
(2.26)
2.3 Box Wavelet
21
By the definition (2.20) of the box wavelets we then infer the two-scale relation i. _ "" (Wi+l,rl)I/2 j,k ~ . (2.27) dJ ,k ,e L.J IV k I a e,r UJ+I,r, e E E*. r EM j ,k
J,
On the other hand, the two-scale relation (2.4) for the cell averages can similarly be written as (2.28)
In matrix-vector form th e equations (2.27) and (2.28) then read cl j ,k
= Aj,k Aj,k Uj+l ,k '
Here the vectors are defined by UJ+I,k := (Uj+l ,r)rEMj ,k and clj,k . (dj ,k ,e) eEE where we use the convention dj ,k,O := Uj ,k ' Since the matrix Aj,k is assumed to be at least invertible in order to realize a change of basis , we can locally reexpress the fine-scale averages by the coars e-scale ones and the details , i.e., ~ A-I A-I dUJ+I ,k
=
j ,k
j ,k
j ,k·
So far, we have only considered a local change of basis. This has been possible, because the support of the box wavelets .f!;j ,k, e , e E E, is completely covered by th e support of the box function rpj,k . According to the univariate case we write the global change of basis in terms of the vectors ~ j and 4"j,e := (.f!;j,k ,eh Elj ' e E E . In particular, we obtain (2.29)
and (2.30)
Here the ent ries of the mask matrices are determined by fh,J .,e _
r ,k -
rEM j ,k } ,elsewhere'
{ aje,T ,k ajO,T' ,k
0
gJ,. e k, r
= { bj,kjaj,k r ,e O,r ' r
0
EM} j,k
, elsewhere
(2.31)
for r E IJ+I, k E I j , e E E, according to (2.22) and (2.24) . Note , that bt:~ = a{:~ provided that the matrix A j ,k is orthogonal. The sparsity pattern of th e mask matrices is only presented for the two-dimensional case
**** Gj ,e
****
=
****
****
22
2 Mult iscale Setting
where we assume that each cell is decomp osed int o four subcells, see Fig. 2.1. The pattern of M j ,e coincides wit h the pat tern of t he t ra nspose of G j ,e ' In particular, t here is exactly one entry in each row and colum n due to t he nestedness of the grid hierarchy. We now gather all mask matrices corresponding to t he box wavelet s of type e E E* in one single block, i.e.,
T hen we can define t he matrices
as in t he univari ate case, see (2.12). Note, that t he change of basis (2.29) and (2.30) impli es that Mj and Gj are inverse. Similar two- scale relations hold for t he L'Xl-normalized counte rparts defined by (2.25) for th e vect ors (2.32) wit h t he diagonal mat rix Vj := diag ((llJJ,k lhE IJ . Then we infer from t he loca l biorthogonality pr operty, see Corollary (1), that t he systems ;jj U UeEE* ljI j ,e and j U U eEE* 'It j ,e are biorthogonal provided that t he matrices A j ,k are orthogo na l. Hence, the projection of any function u E £ 1(D) onto piecewise constants with respect to t he refinement level j can be represe nte d by
Finally, we have to verify that the box wavelets have one vani shin g moment , i.e., t hey are ort hogona l to constant functions. To this end we not e t hat ~ aj,k a j ,k (1 in'+1 ) rv = ( aj,k)T a j ,k .L....t O,r e,r 'r J .r t ss 0 e rEMj .k
holds for all e E E * , k E I j, j E No. Again , we need t hat t he vect ors a~, k , e E E , are at least ort hogonal. According to th e univari at e case we then infer t hat t he det ails decay like 11JJ,kI which becomes sma ller with increasing refinement level j .
2.4 Change of Stable Completion In order to improve t he compression rates we need wavelets wit h bet t er can cellation pro perties in the sense that higher order polynomial moments vanish. In [CDP 96] a systematic ansatz has been pr oposed for t his task. To t his
2.4 Change of St abl e Completi on
23
end, we first not e t hat th e ~onstruction of t he matrix M j ,l is only one way t o complement t he matrix Mj,o t o an invertible matrix Mj . There is a cont inuum ?f completi ons of Mj,o . We call a complet ion (uniformly) sta ble if th e matrix Mj and it s inverse Gj have uniforml y bounded operator norms with respect t o a suitable vector norm. For t he matrices Mj and Gj correspo nding to t he completion by mean s of t he box wavelet it can be pr oven that t hey are stable in any lP-norm, see Sect . B.1.2 for p = 2. Not e, t hat any completion Mj,l of Mj,o characterizes a particular complement of t he spaces of piecewise constants relat ed t o t he refinement levels j and j + 1. In particular , t he correspond ing wavelet basis Pj is stable in the sense th at
holds for constants c, C ind epend ent of j. In the sequel, we will const ruc t anot her stable complet ion Mj,l of Mj,o such that th e corre sponding wavelets have higher ord er vani shin g moments. The start ing point is any composed matrix
where t he matrices Lj,e E R N jx N j , N, := # I j , are uniformly bounded with respect to l l . Then we not e t hat t he mat rices
- - (I L.)
MJ· ·= MJ· .
are inverse, i.e., single blocks
Gj
0 IJ
= Mjl. From t he composed matrices we det ermine t he
and, equivalent ly, Mj,o = Mj,o, (2.33) Gj,o = Gj,o -
L
Lj,e Gj,e,
eEE*
where we employ the block st ructure of Lj, Mj,l and Gj,l with respect t o the different wavelet ty pes e E E *. Since t he matrices Lj,e, e E E* , are assumed to be uniformly bounded, t he matrix Mj,l is st ill a stable complet ion of Mj,o. In terms of t he new complet ion, the basis of modified box wavelets can now be represent ed in terms of th e old wavelet basis Pj, i.e., (2.34)
24
2 Multiscale Setting
Not e, t hat t he basis Ej , e E E* , kEIj , j E{ O, . . . , L - 1}}, (3.1) where E = (EO, . .. , EL )T denot es a sequence of t olerances. Again, in slight abuse of not ati on we will refer to this ind ex set as set of significant details. The correspond ing det ails (dj,k,e)(j,k,e)EDL,. is called the sequence of significant
3.1 Ad aptive Grid and Significan t Det ails
35
details. In analogy, we int rod uce th e sequence of local averages (Uj ,k) (j ,k)E9L,e corres ponding to t he adaptive grid. The adaptive grid det ermined by t he ind ex set 9 L ,e can now be const ructed by mean s of t he ind ex set V L ,e where we apply the following refinement crite rion .
D efinition 6 . (R efin em ent criterion) Let 9 be a nes ted grid hierarchy and V L ,e a set of significan t detai ls. Th en a cell Vj ,k , k E I j , is refine d if and only if there is e E E * such that (j, k , e) E V L ,e '
For lat er use we introduce t he local ind ex sets
he := {k
he := {(k,e ) ;
; (j , k) E 9 L ,d C I j ,
(j , k , e) E V L ,d C I j
x E *,
which correspond to t he indi ces of 9 L ,e and V L ,e on one level j . The ada ptive grid 9 L ,e is now constructed by the following refinement algorit hm. Algorithm 2 . (Refinement Algorithm) 1. Initialize It := 10 ;
2. for j = 0 to L-1 do 1. Initialize It +1 :=
0,
I j- :=
0;
2. A pply th e refinement criterion, i.e ., for (k,e) E J j ,e do Ii := Ii U {k}; Tt l := It +! U Mj, k;
3. Discard the refine d cells, i.e.,
h e :=
nv:
Here t he sets It C t, and I j- C t, can be interpret ed as collect ions of ind ices indicating cells on level j which might be refined on th e next finer level when applying t he refinement criterion and t hose which have been refined , respectively. Wh enever a cell Vj,k is refined, t hen t he indi ces of t he new cells Mj ,k C I H 1 have to be added to It 1 and the index k of t he refined cell has to be removed from It. The resulting grid is an adaptive grid in t he sense of Definition 5. The grid refinement is illust rat ed schematically in Fig. 3.2 where t he refined cells are sha ded. Finally, we introdu ce t he locally refined spaces
B~\
""""' L-....
rEM~,k
ar e det ermined by
IVj ,k I I m- s» Bn IV r ,k j+ l,r
_
-
J+I ,r
""""' Bn L-.... j+l,r'
(4.10)
r E M ~,k
where we have used (2.31) and A j ,k := T/IVj ,kl · So far we have only derived evolut ion equations for the average s and details corresponding to the full grids. The adapt ive FVS is now det ermined by a significantly smaller selection of evolution equat ions corr esponding to n+ 1 . tea apt ive gnid rv t.e . i.e., h danti n+l vj ,k
= V jn,k -
\ "'j ,k
Bn
[. k»
(4.11)
Here, t he ad ap tiv e grid is det ermined by the set of significant det ails V£i1 according to the Algorithm 2. We ar e now facing two fund am ental problems, nam ely, - how to compute the local flux balances B j,k without employing the flux balan ces corre sponding to the cells of the finest grid and - how to det ermine V£i1 without evolving the averages on the globally finest level, applying the mul tiscale transformation and finally thresholding the det ails? In t he following two sub sections we will discuss these questions. 4.1.1 Strategies for Local Flux Evaluation According to the definition of the flux balan ces (4.10) the computation of B j ,k requires all flux bal an ces B2,k' corres ponding t o the cells V L ,k ' C Vj ,k of t he fin est level. However , exploit ing the conservation property of the fluxes (4.6) t he right hand side of (4.10) can be simplified . For this purpose, the boundary part r11is decomposed into smaller par ts r{ I' according to the local ind ex sets ' ,
76
4 Ad aptive Fini t e Volume Scheme
stf := {( k' ,l') ; k' ,l' E Ij' , r {,l l C r i ,t>
VJ I,l1 C VJ,d,
0::;
j ::; jf ::; L.
Then t he num erical fluxes and also the flux balances (4.10) can be rewritten as (4.12) Bj,k = !Ft.11 F{',7
L I
wit h the local num eric al fluxes
L
F{',7 =
F{,',~
(k' ,I')Est:f
=
L
Ff,'JI =
L
F(v2,T ; r
E
F f,,ll)'
(k' ,l1 ) Est: ~
(k' ,I')Es t'}
(4.13) Here, t he flux balanc e Bj,k is compute d by the num eric al fluxes Ff,'JI corresponding t o t he boundar y part r f, ,1' C ri,1 C 8 VJ ,k' All num erical fluxes corres ponding t o r f, ,I' inside the cell VJ ,k cancel each other due t o the conservatio n property (4.10). Thus, (4.12) is preferable to (4.10) with regard t o an efficient computat ion. Fur thermore, we observe that th e computation of t he num erical fluxes Ff,'JI requires t he average s V2,T' r E Ff" I' C Ii., on the fin est level which have to be pr ovided by means of t he local inverse two- scale t ra nsformation (3.21) - (3.23), see Fig. 4.1 (left) . This, in general, inflat es t he complexity by a logarithmic factor depending on the spatial dimension d. In par ti cular , for a dyadi c refinement we obtain # = (2d - 1 )L - j . In ord er t o t reat multidimensional pro blems a chea per approximation is desirabl e. If t he reference FVS is based on num erical fluxes corre sponding to a st ruc t ure d grid, one alte rn at ive is t o make t he grid locally uniform , Le., th e grid is locally refined such t hat t he fluxes can be comp ute d to dat a of the sa me refinement level, see Fig. 4.1 (middl e) . For the computat ion of t he num erical fluxes F{',7 , (j, k) E 92}1 , we have t o distinguish three cases, nam ely,
sk'f
(i) the neighb oring cell VJ,I also belongs to t he ada ptive grid, Le., (j , I) E r.n+l v i ..e » (ii) the coarser cell VJ-l ,11"j(l) belongs to t he ada pt ive grid, i.e., (j -1, 7["](1)) E 92}1 or, equivalent ly, (j - l ,7[j(I) ,e) ~V2il , e E E * , and (iii) the neighboring cell has been refined due t o significant det ails, i.e., (j, I, e) E V£i 1 for some e E E * . Here, t he gra ding of the grid implies t hat t he levels of two neighboring cells differ at most by one. In case (i) and (ii) the num erical flux is computed by (4.14) and in case (iii) by ,n F jk,1 -
'"
L.J
(k',II)Est ',{+l
Fjkl+l ,I',n =
'L.J "
(k
l,l'
)ESU+
F( Vj+l n ,T ;
r
E ,rkl 'T"j +l) ,I"
(4.15)
1
P ro ceeding in such a way, we event ua lly have to compute addit iona l averages V.f+l ,T usin g (3.21) - (3.23). In genera l, the num eric al fluxes do not
V'j,T or
4.1 Construct ion
77
only depend on two values corresponding to the two adjacent cells but on a larger st encil, e.g., high- ord er accurate FVS employing a higher order reconstruct ion . Therefore t he neighborh ood of cells to be made locally uniform is larger. To this end , we have introdu ced t he notion of a locally uniform grid of degree p , see Definition 8. Here, the degree p can be ensur ed if t he grid is graded of degree q where q is chosen sufficient ly large, see Theorem 4. This again imposes a constraint on t he grading paramet er for t he set of significant details, see Proposition 1. Another alte rnative of making t he computation more efficient is t he use of an unstructure d approac h. Here t he num erical fluxes are computed by mean s of t he averages pr ovided by t he ada ptive grid, i.e., (4.16) In t his case no local refinements are necessary, see Fig. 4.1 (right) . Although the two alte rnatives are less expensive th an th e exa ct comput ation according to (4.12) we have to be aware th at an erro r is introduced . As investi gations in [CKMP01] show, this erro r might become significant in case of a first order FVS. However , if t he reference FVS is a higher order scheme employing a high- ord er reconst ruction, t hen no significant loss in accuracy has been observed.
~-
I
' 0 -:-,0
- - I - ....' 0
0 '
• • 1• •
0 :
r- -:-0 .0 .:,.. 0 '
,, ,, 0
---- .,, ---,,
0
0
0 0
0
0
Fig. 4.1. Exact (left ), locally st ructured (middle) and un structured (right) flux eva luation , (.) numerical fluxes, (0) cell averages
4.1.2 Strategies for Prediction of D e t a il s
The core ingredient of the ada pt ive FVS is t he pr edict ion of th e set VZ~l by means of t he dat a {Vj, d (j ,k) E9 r: .€ and {dj,k,J(j,k,e)E'Dr:,€' respectively. Afirst ap proac h was int roduced by Harten in [Har9 4, Har9 5]. It is based on heuri sti c arg ume nt s employing chara ct eristic features of hyp erb olic conservation laws, see also Sect . 1:
78
4 Adaptive Finite Volume Scheme
- The speed of propagation is finite. This implies that information moves according to the locally finite characteristic speeds, i.e., the eigenvalues of the Jacobian corresponding to the normal fluxes (1.3). The time-space continuum limited by the characteristics propagating with maximal and minimal characteristic speeds form the range of influence. For systems of conservation laws this is a cone. - Discontinuities may develop although the initial data are smooth. In case of nonlinear conservation laws this is caused by the intersection of characteristics.
V2;i
by means of From these observations Harten deduced a prediction set two criteria applied to each significant detail (j, k, e) E VZ e of the old time level, namely, ' (i) details in a local neighborhood of a significant detail may also become significant within one time step,i.e.,
Ildj,k,elloo ,* 2: Cj
=>
(j,r,e) E
V2;-i , r E Nj~~ ,
e E E*,
(4.17)
and (ii) gradients may become steeper causing significant details on higher levels, i.e.,
Ildj\,ell oo ,* 2: 2M +1 Cj
=>
(j
+ 1, r, e) E V2-;g1,
r E M1,k' e E E* . (4.18)
In principle, it suffices to choose q' = 1, since the CFL restriction for explicit FVS ensures that the time step is sufficiently small such that a perturbation is propagating only from one cell to another. To guarantee a reliable adaptive scheme the prediction set V2-;g1 has to satisfy the reliability condition
tr:z .e U trr: L,e
C
n 1 Vt..e + :
(4.19)
So far, Harten's approach has not yet been verified to fulfill (4.19) . In [CKMP01] a slightly different strategy is proposed. The basic idea is to determine all details dj';k~ .e' on the new time level which are influenced by a detail dj,k ,e on the old time level. In order to represent this influence set Vj,k, e we have to determine all averages vZ ,r' r E Ej,k,e C It. , that are influenced by the detail dj,k ,e' i.e., the range of influence of dj,k,e ' Accordingly, we determine all averages V2~1, r E tj' ,k' .e' Chon which the detail dj';k~ .e' depends, i.e., the domain of dependence of dj;;k~ .e' In order to merge the index sets Ej,k,e and t j,,k'.e' which correspond to different time levels, we either have to determine the backward influence domain ir: k' , or the J , ,e forward influence domain by means of the evolution equation on level L. In the sequel we describe how to compute the sets Ej ,k,e and tj' ,k' .e'> Since we also need the range of influence and the domain of dependence for
»i;
4.1 Con struction
79
j+1 4(k-s)
4k I
I
4 (k+s)+1 e
2(k-s)
I
e
2k
I
•
I I 2(k+s)+1
J
j-1
e
k
Fig. 4 .2. Illu strati on of the range of influence Ej~k , e
the averages V'J,k we introduce the convention d'J,k ,O := V'J, k'
9;::,
Range of influence. The set E j,k ,e is det ermined by the supports e E E , according to the local invers e multiscale transformation (3.21) (3.23). From these two-scale relations it can be recursively computed U) '= {k} E J,k ,e . , E(lk+l) J , ,e
"
:=
U
(I )
r E E.
J , k, e
._ ,,(L)
L.Jj ,k ,e . -
LJ j,k ,e
91*,e , l = i ,... , L - 1, , 1'
'
9;:: 9;::
Since the supports of the modifi ed box wavelets fulfill = = M1,k' e E E*, and t he grids are nest ed , we may int erpret t he corre sponding range of influence as E j ,k ,e = {r E h ; VL,r C Vi,d, e E E* , i.e., t he partition of the cell Vi,k by cells on the finest level. This do es not hold for e = O. In t his case the support corresponding to t he ind ex set E(lk ) 0' J, , i.e. , U r EE (l) Vi,r:J Vi,k , is successively growing with increasing levell. This J .k ,O
is illustrat ed in Fig. 4.2 for the one-dimensional case according to Sect . 2.5.2. Here we ass ume that the st encil of the modifi ed box wavelets is symmetric, = {2(k- s) , . . . , 2(k+ s)+1} , i.e., J1 = 0 in (2.39). In par ticular , we obtain
9;:2
cr. Sect. 3.8. The stencils EJ~k ,o and EJ~L corr espond t o the cells indicat ed by both. or ° and 0, resp ectively. Domain of dependence. The ind ex set E j ,k ,e , e E E , can be compute d by mean s of the local multiscale t ransformat ion (3.20) and recur sively applying t he two-scale relation (3.16) f', U ) L.J
._ M e
i ,k,e '-
i ,k'
80
4 Ad ap t ive Finit e Volume Scheme 11
1 1 10101 010 18 18 18 18 10 1010\010 101010 101010 1010 10 1010 1 1 1 1 1
8(k-s) 1
0
1
I 8
0
1 8
1
4(k - s)
0
1
0
I
I
0
1
0
0
I
0
I
4k 1
1
2 (k-s)
0
1
1
0
4(k+s)+1 8
8
2k
2k+1
1
1
2 (k+s)+l
F ig . 4 .3. Illustration of t he dom ain of dependence
U
E-(1+1) j k e: = ,
-
- (1)
rEEj, k ,e
1
._
j +3
8 (k+s )+1
8k
-°
M 1+1 r' l '
= j , ... , L -
£?L 2,
(L - 1)
E j,k,e .- E j,k,e
Not e, that this set corresponds t o the support of the modified box wavelet 'l/Jj,k ,e, e E E *, according to Algorithm 1, and t he box funct ion 'Pj,k == 'l/Jj, k,O, respecti vely, i.e., supp 'l/JJo k e = U r EE- J ,k , e VL ' r - Similar t o t he set E j k e i e E E *, we may int erp ret t he dom ain of dependence as °
J
I
I
U {r E t c ; VL,r C VJ+1 ,s},
E j,k,e =
I
e E E,
s E Mj, k
i.e., t he par tition of t he cells VJ+1 ,s, s E M j,k by cells on the finest level. In analogy to the ran ge of influence, we consider again t he one-dimensional case where we employ the modified box wavelet with symmet ric support , d. Sect . 2.5 .2. According to Sect. 3.8 we det ermine M ~,k = {2k , 2k + I} and
M ],k = {2(k - s), 2(k + s) + I} . In Fig. 4.3 t he index sets by
0
(e = 0) and . or
0
EYL are ind icated
(e = 1), respectively.
B a ckwa r d (fo r ward) influence domain . By mean s of the explicit evolut ion equation (4.1) we know how t he averages of two t ime levels are correlat ed . For t he sa ke of simplicity, we assume that the discreti zation st encil which is involved in the computation of the numerical flux balan ce 82 k and the corresponding numerical fluxes F~'t , see (4.2)and (4.3) , is determi~ed by t he ind ex set Nf k indi cating t he neighborhood of degree p corre sponding t o t he cell VL,k on t he finest level, i.e., U, :Ff,1 C Nf ,k' In order t o merge t he ind ex sets E j,k,e and E j , ,k' .e' we have two alte rnat ives based on looking back or forward in t ime. One possibility is t o collect all indi ces corres ponding to averages v2,r ' r E E;; ,k'.e' on t he old time level which are needed when computing v2:~? , r E Ej' ,k ' .e' according t o t he evolution op erator , i.e.,
E-:;J, k ' .e , :=
U
Nfr' ,
4.1 Construction
I~I I
0
I
0
I
0
I
0
I
I
I
0
I
0
I
0
101
I~I
I
0
0
81
Fig. 4.4. Backward (left) and forward (right) influence dom ain
Conversely, we can det ermine t he num erical range of influence corresponding t o t he averages vl,r' r E E j ,k,e , i.e. , E tk ,e :=
{r
E
ti. ; Nf,r n E j ,k,e =l0} .
The different points of view are illustrat ed in Fig. 4.4 . Influence set . The influence set V j ,k,e of det ails dj';k1, .e' which depend on t he det ail d j ,k,e and t he coarse-scale averages v O,k = dO,k,O can be represent ed as V j ,k,e =
{(j', k' , e' ) E V L ,O
; E;; ,k' ,e'
-
= {(j' , k' , e' ) E V L ,O ; E j' ,k' ,e'
n E j ,k,e =l0} n Ej+,k,e =l0}.
To realize an efficient impl ementation we have to det ermine a priori t hose indi ces (j' , k' , e' ), which yield a cont ribut ion to V j ,k,e' We not e t hat the det ail dj,k ,e may not only cause a perturbati on in t he neighborhood of t he cell Vj,k similar to (4.17) but may also influence det ails dn'+k~ e' on higher scales. In cont rast to Harten 's st rategy, here j ' > j + 1 is J, , also admissible. Since th e addit iona l higher levels inflat e the influence set, we would like to bound th e number of higher levels t o a minimum number which st ill pr ovides t he reliability property (4.19). For t his purpose we fix some a > 1 which will depend on th e smoot hness of t he primal wavelet s, see Assump tion 4 in Section 5.3. We now assign t o each significant det ail or coarse-scale average corres ponding to (j , k , e) E V'l,e and (0, k, 0) == (0, k ), respecti vely, a unique ind ex v = v(j , k, e) such th at 2 v (j ,k,e) U
C' J
< lid'j:'.k ;« II 00 , * < 2(v(j ,k,e)+1) c '. J U
(4.20)
Since th e ind ex v(j , k, e) becomes th e sma ller th e lar ger a is, it is convenient to choose a as large as possible. Th en th e modified pr ediction set is det ermined by jjn+ l .- V n U L ,e ' L,e
u
(j ,k,e )E'f5~.e
{(j' , k' , e' ) E Vj ,k,e ; j' ::; j
+ v(j, k , e)}
(4.21)
82
4 Ad aptive Finite Volume Schem e
with D~ ,e := D'L,e U {(O , k,O) ; k E I o}. We would like to remark th at t he definition of DZ+g1 is slightly different to th at in [CKMP01], since we apply no t hreshold on th~ coarse-scale averages. This would destroy th e conservat ion property of the ada pt ive scheme, see Lemm a 5. For a scalar one- dimensional conservat ion law it is possible t o est ima te the details dj';k~ ,e' on t he new t ime level by t he det ails d j,k,e on th e old t ime level, see Sect . 5.3, from which we conclude t he reliability property (4.19). Fin ally, we remark t hat we only apply Har t en's st rategy for our computations present ed in Sect . 7, becau se it is mor e efficient and easier to impl ement . This decision is motivated by t he investigations in [CKMP01] .
4.2 Algorithms: Initial data, Prediction, Fluxes and Evolution In the sequel, we pr esent th e algorit hmic ingredient s of t he adaptive FVS . Four basic routines are needed , nam ely, (i) t he multi scale decomposition of t he initi al dat a , (ii) t he pr ediction sets D2i 1 , (iii) t he computat ion of t he local num erical fluxes and (iv) t he evolution of t he averages corresponding t o t he adaptive grid 92i1 . First of all, we consider the comp utation of th e set D~ e- In principle, we have to perform t he fu ll multiscale decompositi on accord ing t o (2.40). However , this would requir e the computation of all averages on the finest grid. In ord er t o redu ce t he comput ational complexity thi s has to be avoided. Therefore we present an algorit hm t hat pro ceeds levelwise from coarse t o fine scale according t o Algorithm 2. The basic idea is as follows: first we compu t e all averages on levell, i.e., the ada ptive grid is assumed to be th e grid of levell , and det ermine t he det ails by t he local two- scale tran sform ation (3.17) - (3.20) . If there is at least one significant det ail corre sponding to a cell V O,k t hen t his cell has actually to be refined otherwise t he ada pt ive grid is locally coarsened. Aft er having handled all exist ing cells of t his level we th en proceed to t he next high er level where we refine again t he cells of t he locally highest level and det erm ine t he det ails by the local t wo-scale tran sformation. These det ails det ermine which cells act ua lly remain . This pro cess is successively repeate d until th e a priorly fixed highest refinement level is reached . As input dat a th e algorit hm needs (i) t he number of refinement levels L, (ii) th e sequence of t hres hold values e , (iii) t he scaling par am eter t ol and (iv) t he initial function Uo. Then t he algorit hm provides t he set D~ e of significant det ails corres ponding to t he initi al dat a Uo. Note, th at t his s~t is not necessaril y gra ded. Moreover , it is not gua ra ntee d from an analytical point of view t hat this set actually contains all significant det ails . However , in many applications the initi al dat a are given by piecewise constant dat a . In t his case t he set is supposed t o be reliable.
4.2 Algorithms: Initial dat a, Predicti on , Fluxes and Evolution
83
Algorithm 7. (M ultiscale decomposition of ini ti al data) I o+ ..- I.0 for j = 0 t o £-1 do 1.
o, :=
It U U
eE E *
U kE1J+ Lj,k
2. R H 1 := U kEU; M~, k 3. Uj+l ,k := 1V;~ l . k l JV;+l,k ua(x ) dx , k E R H 1
4. Uj,k = A
j,a LJrEMo] ,k m r ' k Uj+1 ,r>
'"
-
A
lj,e 5 . d j,k,e -- '" LJ rEL ~ k r,k Uj,r + A
i.
LJ r E M ~
'"
r,
j,e Uj+1,r , k m r,k -
A
e E E * , k E 1+ J
+1,i := maxkER;+l I(Uj+l,k)il , i = 1, .. . , m 6. U~ 7. for e E E * , kElt do 7.1 for i = 1 to m do if u H 1 < tol -
OO,'l
t hen (dj,k,e)'; := (dj,k,e)i else (dj ,k,e)i := (dj, k, e)du!;};,~ 7.2 Ildj ,k,ell oo, . := Ildj,k,ell oo
7.3 if Ildj ,k,ell oo,*
> Cj t hen (k , e) E he else delete dj ,k,e
8. It+ 1 := U (k,e)EJ; ,e M j,k
F irstl y, we have to initi alize the set It of cells on level 0 for which we have to compute the det ails. In order to det ermine the significant det ails we pr oceed levelwise starting on t he coarsest level. For each level we have to perform the following st eps. First , we det ermine all indi ces of cells on t he locally highest level j which have t o be refined , see ste p 1, in view of the local t wo-scale tran sform ation. To t his end, t he set It has to be inflated , since additi onal averages Uj,k have t o be compute d in order t o perform (3.17) in step 5. Then t he indi ces of the new cells on the next finer level j + 1 and t he corres ponding averages are computed, see st ep 2 and 3. Before performing t he local two- scale decomposition (3.17) - (3.20) in st ep 5, we first have to provide the averages Uj,k not yet det ermined from the data on the finer level, see ste p 4. Similar to Algorithm 4 we perform t he thresholding and det ermine see ste p 6 and 7. By mean s of the significant det ails we deduce the cells of level j + 1 for which we have to det ermine the details in t he next it eration , see step 8. Finally we remark that all averages UO,k' k E l a, have been det ermined when performing t he loop for j = 0, since It = 10 and, hence, Uo = 10 , These data also have t o be provided by t he multiscale decomposition of t he initi al dat a.
»t:
84
4 Adaptive Finite Volume Scheme
In ord er to predict the ada pt ive grid 92i1 on th e next time level we have t o det ermine the set V'Li1 . According to Sect . 4.1.2 two alte rnatives are at hand. Here, we only outline Harten 's st rat egy which has been applied for our computat ions in Sect . 7. As input dat a we need (i) t he number of refinement levels L, (ii) the sets V2,e and :Jj,e ' j = 0, ... , L -1, respectively, of significant det ails and (iii) t he sequences of significant det ails dj,k ,e' (j, k , e) E V2,e corre sponding to t he old time level t n' Then the algorit hm provides V'L i 1 and jj~t l , j = 0, .. . , L - 1, according t o (4.17) and (4.21), respectively.
°
Algorithm 8. (Prediction due to Harten) qn+l .0 J' -- , . . . , L - 1 Vj ,e .- , for j = t o L-1 do
°
1. for (j, k, e) E .J;~e do
1.1 jj~tl
:=
.f;~tl
U {(j , r, e) ; r E
jI~~~e := jj~~~e U {(j 2. dj,k,e := 0,
3. dj+l ,k,e := 0,
+ 1,r, e)
Nj~~' e E E *}
; r E M 1,k' e E E*}
(j, k , e) E .f;~t l \:Jj~e (j
+ 1, k , e) E jj~~~e \:J1+1 ,e
Not e, t hat we ensure :Jj~e C jj~t l in step 1. Next we describ e th e algorit hms for the computation of t he num erical fluxes. Here we have to distinguish between the intern al and t he boundary fluxes, see (4.3) and (4.4) . In both cases we need t he following input dat a: (i) t he number of refinement levels L , (ii) the set Q 2i1 and i j,t 1 , j = 0, . . . , L , respectively, (iii) th e corre sponding local averages v j,k' (j, k) E Q 2i1 , corresponding to t he old time level t« and (iv) for the boundar y fluxes we have t o provide in ad dition th e prescrib ed boundary data. From t his information the numeri cal fluxes corr esponding t o th e flux balan ces Bj,k' (j, k) E Q 2i1 are det ermined . First of all, we consider th e computat ion of th e boundary fluxes. To this end we introduce the ind ex set IJ, j = 0, ... ,L, of t he ghost cells in ord er t o C 8 a, l E IJ. handle t he edges
it,
Algorithm 9. (Bounda ry fluxes) 1 for (j , k) E Q2i , 8 Vj ,k n 8 n # 0 do for 1 E
IJ, rl,l c 8 n do
j ,L do 1. -for (k' , l') E S k,l -
4.2 Algorithms: Initial dat a , Prediction , Fluxes and Evolution
85
1.1 det ermin e th e state v k ' ,I' from th e bounda ry values or th e attached flo w field
,n 2 . Fjk,1
'"
:= L.J (k' ,1' ) E St·~
F L ,n
k' ,I'
For the boundar y fluxes appropriate st ates V k' ,I' have to be det ermined from t he boundary condit ions or the at tached flow field . In t he latter case, we event ua lly need t he average s on the finest level. Then we have to apply the local inverse two- scale tran sformation (3.21) - (3.23) recur sively. According t o (4.4) t he boundar y fluxes corres pond to the fin est level in ord er to avoid inconsist ency. This is a serious dem and , in par ticular , if the boundary {J fl is cur ved. According t o Sect . 4.1.1 there are th ree alte rnatives for the computat ion of t he int ernal fluxes. Algorithm 10. (Exa ct flu x evalu ati on) for j = L downto 0 do for k E i j,t 1 do for l E t . , r il =f. 0, r i l n {J fl = 0 do 1. -for (k f , If) ~ S kj ,L do ' ,l1.1 reconstruct th e states V'l,T' r E F f,,1 " by me ans of recursively apply ing th e local inverse two -scale transformation (3.21) - (3.23), if not yet available
1.2
Ff,. ), = F(V'l,T ;
r E Ff, ,1' )
,n '" F L ,n 2 . Fjk,1 := L.J (k' , I ' ) E st·. ~ k' ,I'
Here, we pro ceed from fine to coarse. For each cell Vi,k of the ada ptive grid we det ermine the num erical fluxes at t he int erfaces to th e neighboring
it,
cells
Vi,I . According t o (4.12) t he num erical flux Fk',7 is the sum of num erical
fluxes corres ponding t o t he partition r l.l = U (k' ,1' )Est.~ rf.; ,I', see (4.13) . The computat ion of t he int ernal fluxes Ff,. )" i.e., r f.;,1' · C fl , depend s on t he averages vt. T' r E F f, I' C lt. , on the finest level. If th ese dat a are not available, i.e., (L ,r) ~ i 211 , th ese values have to be reconstructed locally. For t he boundar y fluxes appropria te states Vk ' ,I' have to be det ermined from t he boundary conditions or the attached flow field. In order to avoid t he local reconst ru ction of valu es on the full fin est level, t he second st ra tegy is based on t he idea t o make the grid locally uniform. Then we can st ill use a num erical flux based on a structured grid , see (4.14) and (4.15) . In addit ion t o t he above input dat a we need t hat the ada pt ive
86
4 Adaptive Finite Volume Scheme
grid can be made locally uniform of degree p where p is determined by the minimal degree q of neighborhoods Nj~k such that UI Fk,1 C Nj~k'
Algorithm 11. (Locally structured flux evaluation) for j = L downto 0 do for k E ij,t 1 do for l E I j ,
rt. 1- 0, r{1 n 0,
where uL denotes th e average of th e exact soluti on , th en th e accuracy is preserved by th e adaptive sch em e provided that e ,...., 2 - (1+0 ) L.
5.2 Stability of Approximation
93
Proof. Since for an explicit FVS t he t ime ste p T is restrict ed by t he CFL condition, i.e., T is proportional to t he maximal diam et er of th e cells, here 2- L , t he assertion follows by Theorem 5. 0 The usefulness of t he above theorem crucially depends on t he verification of t he assumptions (AI) - (AS). First of all, we remark t hat for scalar conservat ion laws on D = Rd t here are severa l first order accurate schemes which are iI- cont ractiv e, so-called monot one schemes, see e.g. [CM80]. Fur th ermore, we noti ce t hat (A2) is satisfied if t he local flux evaluation is performed according to (4.13) . In t his case no error is introduced , i.e., C 2 = O. In case of (4.14), (4.15) and (4.16) , respectively, we introduce an erro r, bu t so far t his err or has not been verified to be bounded by € . Concerning t he consiste nt discreti zation of the initi al dat a a natural choice is given by t he approximat ion opera t or , i.e., '1 := A v o v1. In this case, (AS ) holds pr ovided the c.e approxima t ion error is uniformly bounded in the sense of (AI). We emphasize t hat the application of th e approximat ion operator in general requires O(Nd opera t ions. In pr acti ce, a cheap er strategy is pr eferabl e, see Section 4.2. In the following sections we now address to some exte nt t he st ability of the approximation operat or and t he reliabil ity of t he pr edictiv e set of significant details.
5.2 Stability of Approximation In order to verify t he stability of t he approximation erro r it is convenient to investigat e th e convergence of t he so-called su bdivisi on algorithms. For t his purpose, we rewrit e t he inverse local multis cale t ra nsformation (3.21) (3.23) in full matrix- vect or representation
Recursively applying t his two-scale relat ion yields L- l
UL = G{;,D UD
+L
L
G1,eclj,e,
(5.4)
j=D eEE '
where th e matrix G1,e E RNL X n, is defined by (5.5)
represents t he subdivisi on procedure. Next we introdu ce t he kth column of G1,e which can be regar ded as discrete basis vect ors ,T.L ._ ,T.L C ~ j,k ,e . - ~ j ,e j,k
(5.6)
94
5 Error Analysis
by means of the Dirac vector Cj,k := (Ok ,r)rEf;, see [Dah94]. Then we can rewrite (5.4) as
UL =
L
L-l
'lJ6',k,oUO ,k +
kEfo
L L L j=O
eE E *
'lJ1,k,e dj,k ,e.
(5.7)
»ei,
Here we only consider a scalar problem. In case of a system of conservation laws we can apply the scalar investigations componentwise. Consequently, the approximat ion error can be written in the form
L
UL - Av UL =
'lJ1,k ,e dj ,k,e.
(5.8)
(j,k ,e)f/.V
In order to control the threshold error we need the subdivision scheme to converge at least in the ll- met ric, i.e., the piecewise constant function 'l/J1,k,e defined by (5.9) converges to a function 'l/Jj,k,e in t», It can be proved that the convergence of the subdivision scheme corresponds to the existence of a biorthogonal wavelet system. For general surveys on subdivision algorithms we refer to [Dyn92 , CDM91] and to [Dau92, CobOO] for their relations to wavelets. So far, results are only available in case of uniform refinements on structured grids, see e.g. [CDF92, DKU99, CDKPOO] . In case of curvilinear grids or arbitrary grids no stability results are available yet . In the sequel, we employ some standard results from the theory of subdivision schemes which shall be summarized for the convenience of the reader. For this purpose we first introduce the notion of quasi-uniform grids. Definition 9. (Quasi-uniform Grid) L et 9 be a nested grid hiera rchy . Then the grid on level j is called quasiuniform if there are constants a < c ::; C such that cT j < diam(Vj,k) ::; CTj and cTjd::; IVj,kl ::; C2- jd for all positions k E I j and levels j = 0, . . . , L ,.
Then we obtain for the primal wavelets the following results.
Proposition 3. Assume that the piecewise constant functions 'l/J1,k,e defined by 'l/J1,k,e(X) := ('lJ1,k, e)r ,
E VL,r, r E
X
h
(5.9)
conv erge uniformly in L towards a function 'l/Jj,k,e E L OO(f!) in the sup -norm. Then th e limit functions (primal waveletsy satisfy the following prope rties: 1.) Any funct ion u E L OO(f!) can be un iquely expanded in a series of the primal wavelet basis, i. e.,
u= 1
The functions
I:
kEfo
1jJj,k,O
(u, O' Consequently, we can estimate the supremum of vI. by a constant only depending on T and the supremum of the initial data uo .
o So far we have not verified the reliability property (4.19) for the prediction set DZi1 . However, we may assume that (4.19) holds for all previous time steps J = 0, ... , n in order to check the reliability of the prediction set for the new time step. As we will see in the proof of Proposition 5, it will be sufficient to know that the data of the old time step are uniformly bounded by the constant C(T, uo) . Proposition 5. (Finite differences for composite functions) Let the assumptions of Lemma 8 hold and assume that the flux balance function B satisfies Assumption 3. Introducing
DN(vI., K,
i\d
I(R) := {(j,k)
:=
j
{ILl% vI.,k,1
sup
; S(N, K, k')
c Ej 'k}
j E {l ,oo. ,R}R, k E {O, .. . ,R}R,
L:
and
1
j1k1 =
R},
we obtain
D R(BI.,K,E j,k):::; C sup
{II: (Dj/(vI.,K,E;'k))k/ 1
(j,k) E I(R)} . (5.22)
Proof. We want to estimate the finite differences Ll~ B2 k by means of finite differences of possibly lower order. To this end, we first 'introduce Lagrange polynomials Pi E PR, i = 0, ... , 2p, of degree R defined by the interpolation conditions (5.23) Pi(m) = vI.,k-p+i+mK' m = 0, ... , R ,
and the composite function G : R -7 R G(x) := B(po(x), .. . ,P2p(X)) == B(P(x)) .
The definition of the finite differences implies
Ll~ B2,k =
R
l; (_l)m (~) G(m)
=: Llf G(O)
and thus we deduce from standard results for finite differences, see e.g. [SB80, DL93],
ILl~B2,kl:::;C
sup IG(R)(x)l·
(5.24)
xE[O ,R]
Furthermore the R-th derivative of the composite function can be represented by
5.3 Reliability of Prediction
105
where we employ the smoothness of t he flux balance function B according to Assumption 3. When P(x) E {)D i , we consider th e onesided continuous ext ensions of the derivatives. This representation can be derived successively applying the chain rule for differentiation. Then we can estimate the R-th derivative by sup
IC(R)(x)l:::; C sup
xE[O,R]
xE [O,R ]
{II~ Ip;:d(xWI ;
where t he constant C depends on the coefficients and the bounds sup {{) xE[O,R ]
(j,k) E I(R)} ,
(5.25)
1-1
Cj ,i ,k
and R, respectiv ely,
()k B{) (P( x)) : i E {O, .. . , 2P}k } . Pi! . .. Pik
From the Lagrangian repres entation of the polynomials Pi we conclude that there exist s a uniform bound such that sup
IPi(X)I:::;Cllv2111=,
i=0 , . .. , 2p.
x E [O,R]
According to th e assumptions and Lemm a 8 we know that the reference FVS is uniformly bounded in the sup-norm. Hence, the bounds Ck only depend on T and IluoIIL=. We now consider the Newton representation of t he int erpolation polynomials Pi , i.e., R
Pi(X) =
v -1
L:>dO, . .. , II] II (x v =O
1=0
l) =
R
v-1
v=O
1=0
L ~II. L\r Pi(O) II(x -l) ,
(5.26)
where pdQ , . .. , II] denotes the II- th divid ed difference, see e.g. [SB80, DL93]. By means of induction and using the addition theorem for binomial coefficients we notic e th at
Incorporating t his relation in (5.26) we dedu ce sup
x E [O,R ]
Ip;j)(x)! :::; C _ max .1L\{Pi(l)l , j:::; R ,
(5.27)
I-O ,...,R-J
wher e C only depends on R. Not e, that t he t erms for II = 0, ... , j - 1 in (5.26) give no contribution to t he j th derivative. Furthermore th e definition of t he finite differences and the int erpol ation conditions (5.23) yield
106
5 Error Analysis (5.28)
Combining (5.24) , (5.25), (5.27) and (5.28) we obt ain
(i, v) E I(R ,p,j)} , where I(R,p ,j) := {(i,v) ; iE {O, .. . , 2p}R, VI E {l , ... , R - j d R } . Note, t hat t he bound C only depend s on R and Ci : We now esti mate t he finit e differences on t he right hand side by
ILl}( VI: ,k-p+H vl K I ::; D j l (vI:, K , Ej,k) ' where we take into account t hat the set Ej,k is inflated by the ste ncil of the numerical fluxes specified by the par am et er p. Therefore we need E~k inst ead J, of E j,k. Finally we obtain the asse rt ion taking the supremum over all k' such t hat S (R , K , k') C Ej,k' D Next we deriv e a converse result to Proposition 3, i.e., we want to est imat e t he finit e differences LlW U L ,k by certain det ails dj,T' For this purpose we employ an inverse (Bernstein-typ e) est imate which is a standa rd t ool in approxim ation t heory. Lemma 9. (In verse estimate) Let 1::; p,p' ::; 00 such that l/p+ l/p' = 1. As sum e that the scaling junctions CPj,k and uvector typedef tvector_n< double, 2-d-1> evector typedef tvector_n< uvector, 2-d-1> dvector from the te mplate class tvector_n by which we can st ore the vect or of conservativ e qu antities v j,k E R'" , th e det ails dj,k,e E Rm and the vector of det ails corres ponding t o all wavelet types (dj,k,e)eEE• . In the cur vilinear case, t he average s and the det ails ar e enumerated by multiindices k = (k 1 , • •• , kd ) E Ngand level-multiindices (j , k). To this end , we derive the int eger arrays typedef tmultiindex< d > mi typedef tlevelmultiindex< mi > lmi from t he t emplate classes tmultiindex and tlevelmultiindex, respecti vely. For the local t ransformat ions we always access simultaneously all nonvani shin g elements of a column of t he mask matrices. Therefore it is convenient to agg lomerate t he indic es of t he corre sponding supports in one dat a st ructure. However, we do not have to st ore ind ex by ind ex, since in the curvilinear case t he supports can be repr esent ed as a mul tidimension al integer int erval k + [0, i ]d C Ng charac te rized by t he multiindex k and the number i , By this knowledge we designed t he special te mplate class tmultirange. According t o the supports of t he box function and the modified box wavelet s, see Sub secti on 3.8, we need four different typ es of mul tiran ges typedef tmultirange< 1, mi> mr1 typedef tmultirange< 2, mi> mr2 typedef tmultirange< 4*8+2, mi> mr48 typedef tmultirange< 2*8+1, mi> mr28 for t he supports M;:~ (9J,k), M~,k (9;:~) , M j,k and £ j ,k' resp ectively. Here the first argument specifies the length i and the second argument the initial index k. Analogously, we can sto re t he non -van ishing matrix elements corres ponding to t he columns of the mask matrices by multiran ges where in addit ion to the blocked multiindi ces the corre sponding matrix element s are st ored whose t yp e is specified by the third argument . Here we distinguish between t he t ypes typedef tmultirange< 1 , mi, double >mrA_GOc typedef tmultirange< 2, mi, double >mrA_MOc for t he matrix columns of Gj,o and Mj,o , respect ively. In case of t he matrices LJ·, e, e E E*, we pr oceed differentl y. By construc t ion the supports £'7 J , k are
6.3 Data Structures
119
the same for all wavelet types e E E* , only the matrix elements l~'~ dep end on the wavelet type. Moreover , in the local tran sformations we al~ays have t o acce ss all ty pes corres ponding to an ind ex pair (j , k) . Therefore it is mor e convenient to st ore all matrix elements l~:~ , e E E* , in one multirange typedef tmultirange< 2*s+1, mi, evector > mrA_L. In t his we avoid t he multiple st orin g of the ind ex k and the length i of t he int eger int erval. The mask matrices Mj ,o , Gj,o and Lj = (Lj,e)eEE* are then st ored in hash map s, see Sect. 6.2, where each element is a multiran ge representing the nonvanishing matrix element s corre sponding to one matrix column typedef thashmap_linked< lmi, mrA_MOc > liMap_MOc typedef thashmap_linked< lmi, mrA_GOc > liMap_GOc typedef thashmap_linked< lmi, mrA_L > liMap_L Here the first argument of the hash map represent s the type of the key, e.g. a level-multiindex , and the second argument t he type of t he valu e, e.g. a mul ti ran ge. Analogously t he local averages V j ,k , (j , k) E 9L,e , and t he significant det ails d j ,k ,e , (j ,k,e) E V L,e , are st ored in linked hash map s typedef thashmap_linked< lmi, uvector > liMap_u typedef thashmap_linked< lmi, dvector > liMap_d , where each element is eit her a vector of conservative quan ti ties or an array of vectors representing the det ails of all conservat ive qu antities and all wavelet ty pes . Note, t hat t he ind ex sets I j,e and :/j,e are implicitly det ermined by t he linked list s corres ponding to the different levels. The local tran sform ations are perform ed levelwise. To t his end, we need dat a structures where t emp orar y dat a corres pond ing to one level can be stored . Again we use hash map s. Since only on e level is involved , we can simplify this ty pe of hash map. Therefore the dat a ty pes typedef thashmap_linked_one<mi > iSet typedef thashmap_linked_one<mi , uvector > iMap_u typedef thashmap_linked_one<mi, dvector > iMap_w use t he te mplate class thashmap_linked_one. Note, t hat t he hash map has t o be linked , since we have t o traverse throu gh all elements. In particular , t he dat a type iSet is a hash map where only keys are st ored bu t no valu es. This can be int erp ret ed as a set of keys, e.g . mul tiindices. Finally we need a data structure for the man agement of t he ada pt ive grid and corres ponding geometric inform ati on . For inst an ce, t o each cell volume, mono mials of higher order, norm als, arc lengths etc . have t o be st ored . Which information are needed does not only depend on t he mul tiscale set t ing bu t also on the FVS. We therefore do not specify the det ails bu t collect t he geomet ric dat a in a class of its own class cell_data { ... }. Then t he adaptive grid is stored in the hash map ty pe
120
6 Data St ru ctures and Memory Management
typedef thashmap_linked liMap_grid where each element is an element of th e class cell_data. By means of t he above dat a structures we now introduce the hash maps MOc liMap_MOc liMap_GOc GOc liMap_L L for t he management of th e mask matrices Mj,o, Gj,o and Lj , j = 0, . . . , L - 1, as well as t he local averages and th e significant det ails liMap_u v_map liMap_d d_map corresponding t o t he adapt ive grid and th e set of significant det ails. The adaptive grid and related geomet ric inform ation are st ored in t he hash map liMap_grid adapgrid_map where not only t he locally finest cells are taken int o account but also all cells on coarser levels. The initi alization of t hese hash maps crucially influences t he performan ce of t he computation. We th erefore describ e t he choice of nMax , see Subsect ion 6.2, which determines t he length of the hash table and the size of the memory needed for st oring nMax elements. First of all, we consider the mask matrices. We observe t hat t he number of elements corresponds to the number of columns of t hese matrices, since each element in the hash map represents the non-vani shing matrix coefficients corresponding to one column. For the matrices Mj,o and Lj th e t ot al number of columns for all levels is det ermined by ~ 1_ qL
c: N j
=
N L- l
l-=-q
j=O
since N , = q N j H
.
In case of the matrix Gj,o we obtain the total number L 1 _ qL+l "" N , = N L - - ~ j=O
1-q
This is also an upp er bound for t he hash map adapgrid, since we st ore not only cell information corres ponding to the ada pt ive grid bu t also the information for all coarser cells. For th e avera ges and th e det ails the to t al num ber of elements is restricted t o N L for the full grid on t he finest level and accordingly all det ails are significant , i.e., NL - l (1 - qL)j(l - q). From t he t otal number of elements t hat have to be stored in the worst case of a un iform refinement over all levels, we determine t he number nMax = nTotal * rFill of pr edict ed elements where rFill denot es t he fill rat e. Not e, t hat rFill differs from t he fill rat e dFill of t he hash t abl e. In Sect . 7.1.4 we will invest igat e the influence of rFill and nFactor on th e performan ce of t he computation. Finally we would like to remark t hat the hash map s for t he mask matrices as well as t he adapti ve grid are initialized once and then t he length of t he
6.3 Dat a Structures
121
hash table as well as t he memory heap size remain un chan ged t hroughout the computat ion. Therefore t he choice of rFill is mor e significant for the mask matrices and t he adaptive grid. In particular, t he corres ponding hash maps require t he bulk of memor y. This is different for t he hash maps in which t he local averages and the det ails are st ored, since the ada pt ive grid and t he set of significant det ails, respectively, are changing from one time step t o t he next one. Here t he length of th e hash t abl e has to be adapte d in each t ime st ep and t he memor y event ually has t o be exte nded dyn amic ally. From Algorithm 2 we conclude that t he complexity of th e sets 9L,e and DL,e are relat ed by #9L,e = #DL,e , since a cell is refined as long as t here exists a significant det ail. Then we can adapt the size of the hash maps v _map and d_map t o t he actual number of elements det ermined by one of the hash maps with nMax = # DL,e for v _map and nMax = q# 9L,e for d_map , respect ively.
«:
«:
7 Numerical Experiments
In the previous chapters we have developed and investigated a new concept for the construction of an adaptive FVS where we apply local multiscale techniques to a reference scheme. For scalar conservation laws we have been able to derive an error estimate of the form (5.1). This is based on an a priori error estimate for the discretization error of the reference FVS and the stability of the perturbation error in the sense of (5.3) . In [CKMP01] parameter studies have been presented for scalar one-dimensional problems which confirm the analytical results in Chapter 4. In the sequel, we verify that the adaptive concept can also be applied to systems of conservation laws. In particular, we are interested in applications to real-world problems arising from problems in engineering. For this purpose, we present several computations for the two-dimensional Euler equations for a polytropic gas with "( = 1.4, see Example 3. These have been carried out on PC's with a 600 MHz processor (Pentium III) . The numerical investigations are divided into two parts. In the first part we are concerned with the computational complexity of the adaptive scheme with respect to computational costs and memory requirements as well as the stability of the perturbation error. To this end, we present several parameter studies for three test configurations, namely, (i) shock reflection, (ii) implosion and (iii) wave interaction. These are instationary problems performed on Cartesian grids. In the second part we report on some computations for realworld applications. Here we consider (i) the flow over a bump and (ii) the flow around a profile of an airfoil. Both configurations result in steady state solutions. Since the geometry of the computational domains is no longer that simple as before, we discretize the flow fields by block-structured curvilinear grids .
7.1 Parameter Studies The analytical setting of Theorem 5 is much too restrictive with regard to realistic problems. Therefore we are interested in the performance of the adaptive scheme applied to an extended setting not necessarily covered by the analytical investigations. Here three issues are of special interest, namely, (i) the detection of all kinds of singularities, e.g. , discontinuities and kinks,
124
7 Numerica l Experim ent s
corresponding to physically relevant effects by means of the multiscale decomposit ion, (ii) t he complexity of th e adaptive FVS with respect to computationa l costs and memory requir ement s in comparison to the reference FVS and (iii) th e st ability of the perturbation err or. These issues are investi gated by means of several par ameter st udies with respect to an increasing number of refinement levels where th e threshold value e is fixed for all computations. Not e, t hat this does not quite agree with th e ideal strat egy as outlined in t he beginning of Chapter 5. In ord er to balance the discretization err or and t he perturbation erro r , t he th reshold value c has to depend on the numb er of refinement levels L according to (5.1) and (5.2). However , for systems of conservation laws no error estim ate is available so far for the discretiz ati on error. On t he other hand , th e st ability of the perturbation error in the sense of (5.3) has only been verified for scalar problems . Therefore in the following we focus in our num erical investi gations on the stability of t he perturbation error, in par ticular , with regard to its dependence on t he highest resolution level L. Fir st of all, we summariz e t he setting of t he test configurat ions. Then we specify the discreti zation and present th e reference FVS. By mean s of par amet er st udies we investigate and discuss t he computationa l complexity of t he ada pt ive FVS as well as the stability of the perturbation err or for an increasing number of refinement levels L. We conclud e with some par amet er st udies concerning t he parameters rFill and nFactor of the und erlying dat a structures. 7.1.1 Test Configurations We consider t hree configurations where t he computationa l complexity of t he problem strongly varies due to t he singularit ies occurring in the solut ion. Shock Reflection. Thi s is a quasi-one-dimensional problem where we impose a single shock aligned with one of t he coordinate axes, see Figure 7.1. The shock is moving to t he right approaching a wall. At the wall t he shock is reflect ed and moves back to th e left. Not e, t hat th e initial dat a in front of
UL
f--.-
UR
Units St ate L St at e R 1.3 0.8
!2
kg/rn"
Ux
1300
0
uy
mls mls
0
0
p
hPa
11050
5200
Fig. 7.1. Initial configurati on for shock reflection
7.1 Param eter Studies
125
the shock corres pond to a sup ersonic state. Implosion Problem. The initial configur ation is det ermined by two st at es and U/ as well as t he radius r of a circle, see Figure 7.2. Inside t he circular domain we imp ose low pr essure and out side high pr essur e. Again the resulting flow field is quasi-onedim ension al due t o t he rotation al symmetry. With pr opagating time t hree waves develop , nam ely, (i) a rar efaction wave, (ii) a contact sur face and (iii) a shock wave. The shock wave and the contact sur face are movin g t owards t he cente r of t he circle. The rarefaction wave is movin g into the opp osite direct ion where the corres ponding rarefaction fan is expanding. Here we are int erest ed in t he inst anc e when the sho ck wave focuses in the cente r .
UE
Unit s St ate I St at e E
o
e kg/rn" U
m/s
p
Pa
T
m
E
1.251
2.502
0
0
101280 202560 0.15
Fig. 7.2 . Initial configuration for impl osion pr oblem
Wave Interaction. The initial configurat ion is det ermined by four states corres ponding t o th e four quadrants of t he coordinate syste m , see Figure 7.3. Away from th e origin of th e coordinate syste m, the solutio n exhibits a one- dimensional wave pat t ern consist ing of a rar efaction wave, a contact sur face and a shock wave. This structure is obtained by t he solut ion of a one-dimensional Riemann problem characte rized by t he const ant states U L = u j , UR = uj/ and U L = U/ t . U R = U/ , respectively. Close t o t he origin t he different one- dimensional waves int eract formin g a genuinely twodim ension al wave pat t ern.
State I St at e II II
I
I
II
e
0.125
1.000
Ux
1.000
0.000
u y 1.000
0.000
0.100
1.000
P
Fig. 7.3. Initi al configurat ion for wave interaction
126
7 Numerical Experiments
7.1.2 Discretization As the reference FVS we apply an essent ially non-oscillatory (ENO) scheme described in [Mii193] . This E NO scheme is charac te rized by a one- dimensional second ord er accurate reconstruction t echnique via primitive fun ctions, see [HEOC87] . Here the primitive variabl es are reconstruct ed by mean s of piecewise linear pol ynomials. At the cell int erfaces one-dimensional Riemann problems in normal dire ction are solved applying Roe's approxima te Riemann solver , [Roe81]. For all three configurations t he boundary condit ions are simpl e. In principle, three ty pes of boundar y condit ions occur , namely, (i) free stream condit ions at inflow boundaries, see t he left side boundar y in Figur e 7.1, (ii) slip con diti ons at a wall, see t he right side boundar y in Figure 7.1 and (iii) attached flo w fi eld whenever the flow field at t he boundar y is constant in norm al direction. The discreti zat ion of the computat ional domain is summa rized in Tabl e 7.1. Since we employ an explicit FVS, th e st ep size 7 in t ime has to satisfy a CFL condi tion for t he fin est discre tization level L , i.e., AL ,k
max {IAk(U, n)1
; U E D , IInl12=
I} :S c
< 1.
n:=1
h j,i = h j and h j,i Note, t hat for a uniform discretization IVj,k I = 2- 1 h j - 1 ,i = 2- j hO,i . For t he par amet er studies we successively increase t he number of refinement levels L . In order t o maintain the CFL number we have to redu ce 7 by the factor 0.5 when adding an addit iona l refinement level, i.e., 7 == 7L = 2- 17L_l = ... = 2-L70 . From t his we conclude AL ,k = 2 (d- l ) L 70 / h o. Obviously, t he ste p size 7 is charac te rized by the number of refinement levels L and th e ratio 70/ho of th e coarsest discretiz ation list ed in Tabl e 7.1. Table 7.1. Discreti zation pa ram eters rclh« [s/m 2] Reflection [0,1] x [0,0.4] 3.2 x 10- 4 1.28 X 10- 4 10- 6 [0,1]2 3.6125 X 10- 4 Implosion 1.5 x 10- 1 Interaction [-0.4 , 0.4]2 2.5 Case
Q[m]
T [s]
The multis cale analysis is based on t he modified box wavelets. These are const ruc te d according t o Algorithm 1 where t he degree of vani shin g mom ents is chosen as M = 3. In t he cur vilinea r case , the st encils £ j ,k are det ermined by Lj,k with s = 1 according to (3.45) and (3.46). The sequence e of threshold valu es is recursively det ermined by e i. = C, Cj = 0.5 cj+l , j = L -1 , .. . , 0, according to P roposition 3. Here th e threshold valu e is C = 0.001 (implosion,
7.1 P ar amet er Studies
127
wave interaction) and e = 0.01 (shock reflection) for all computations of the par am et er st udy. Fin ally, we have to specify t he flux evaluation. Here we use t he locally st ructure d strategy (4.15) because t he reference num erical flux is only defined on a struc ture d grid. For this purpose, t he adaptive grid has to be locally uniform of degree p = 3 according to the flux stencil. From Corollar y 5 we conclude t hat t his is gua ra ntee d if the tree of significant det ails is gra ded of degree q = 2. 7.1.3 Computational Complexity and Stability
For t he t hree t est configurations we have performed par amet er studies with respect to an increasing number of refinement levels where t he t hreshold valu e € is fixed for all comput at ions. In Figures A.l , A.2, A.5, A.6, A.9, A.I0 t he density is pres ented for th e initial data t = 0 and th e time t = T listed in Tabl e 7.1. The ad aptive grids are plotted in Figures A.3, A.4, A.7, A.8, A.ll , A.12. All these figur es reflect the comput at ional results for t he high est refinement levels list ed in t he Tables 7.2, 7.3, 7.4. We observe that the high er resolution levels are only accessed near discontinuities. This verifies t ha t t he ada ptation crite rion based on det ails is able to detect t he relevant physical effects in t he flow field . We are now int erested in t he computationa l complexity of our scheme (wit h regard to comput ational t ime and memory) for t he prese nt st udies of stability. This is document ed in t he Tabl es 7.2, 7.3, 7.4. The quanti ties list ed in the different columns are - t he number of refinement levels L , - the number N L of all cells corres ponding to the full grid on level L , - t he numb er N 9 := # 9L ,c / N L repr esenting the number of cells corres ponding to t he lar gest adaptive grid determined during t he computation relative to t he full grid on level L , - t he comput at ional ti me
CMS
used by t he ad ap tive scheme,
- th e memory size M em which corresponds to t he maximum amount of memor y that has been allocated in one time st ep, - th e speedup rates SM S := CF VS / C MS det ermined by th e ratio of the computat iona l times for the ada ptive scheme and t he reference FVS on t he full grid of level L and - the per turbation error e L := '1£ - V L E R N L det ermined by t he difference of t he averages '1£ := {vL ,d kEh and VL := {VL ,khEh pr oduced by t he ada ptive scheme and t he reference FVS on t he full grid of level L , respectively. The local averages from t he adaptive scheme are mapped onto the full grid by means of local reconstruction according to t he inverse mult iscale transformation.
128
7 Numerical Experim ents Table 7.2. Param et er st udy for shock reflection
NL
Ng
C MS
%
[min]
M em [MB]
S MS
2560 31 1.50E-1
0.9
2.1
1E-3
10240 16 6.84E-1 40960 83.03E+0 6 163840 4 1.36E+ 1
1.8 3.6
3.7
6E-4 4E - 4
L
3 4
5
7 655360 82621440
2 5.98E+1 13.04E+2
IleLlk L,.
6.8 7.3 12.6
2E-4
14.5 21.8
1E - 4
29.0 34.3
Table 7.3. P arameter st udy for implosion
NL
L
Ng
C MS
%
[min]
M em [MB]
S MS
IleLil1 ,L,.
16384 45 6.833E-1
8 1.34
1E-3
65536 28 3.667E+0 6 262144 14 1.603E+1 7 1048576 76.1l2E+1 8 4194304 42 .194E+2
20 2.10
6E-4
40 3.61
4E-4
84 7.73
4E-4
165 16.51
916777216
334 36.44
4 5
2 7.470E+2
Table 7.4. P ar am et er study for wave inter action
L 2 3
NL
Ng
C MS
%
[min]
6400 89 0.73E+0 25600 58 4.53E+0
4 102400 34 2.33E+1
M em [MB]
S MS
6 0.82 1.2E-3 16 0.98 l.lE-3 39 1.62 1.2E-3 90 2.67 1.2E-3
5 409600 19 1.12E+2 61638400 10 4.99E+ 2
186 4.79
5 1.98E+3
337 9.67
7 6553600
IleL111 ,L,.
Computational complexity. Considerin g t he three tables we make t he following observations: - The number N L increases by a factor of 2 d becau se t he grid is uniformly refined by subdividing a cell into 2d congru ent subcells. - The relative numb er Ng decreases asympt otically by a factor TL := N 9 ,L/N g,L-1' Therefore t he tot al numb er of cells in th e adapt ive grid increases by a factor qt. := # YL ,c:/# YL-l ,c: = 2d TL . In general, t he ratio
7.1 Param eter St udies
129
r L is sma ller than 0.5 and , hence, t he number of cells in the ada pt ive grid increases only by a fact or of 2 at most with each additional refinement level instead of 2d for t he uniform refinement. This results in an exponent ial redu ction of cells in comparison to t he full grid t hat corresponds to significant speed up rat es. However , we would like to emphas ize t hat t he full grid may give us t he more accurate approximation.
- T he computational t ime CMS increases with L -+ 00 becau se t he number of ti me steps is doubl ed according to t he CFL condition for the finest level. On t he ot her hand , t he numb er of cells in the ada ptive grid increases by t he factor qt.. Therefore CMS increases by t he factor 2 qt. which is small er t han t he 2 d +l for t he reference FVS on t he finest level. This results in an expo nent ial increase of t he speedup rates SMS, i.e., t he comput at ional t ime becomes significantl y sma ller for the ada ptive scheme in comparison with the computation of th e reference FVS on the full grid. - Since the number of cells in th e ada pt ive grid increases by t he factor qL , we have to allocate mor e memory space. This is reflect ed by t he quantity M em list ed in t he tables. Comp aring t hese numbers we conclude th at memor y size is increasing by a fact or proportional t o qt.- We emphasi ze th at for Ca rtesian grids th e mask coefficients do not depend on th e level and the position except for boundar y ada ptations and , t herefore, require much less memory. Addit ionally, t hese valu es can be computed in a pr eprocessing st ep. This is not taken into account here. In pr inciple, t hese observa t ions hold for all the t hree test configurations although t he num bers differ according to t he wave st ructures developin g in t he different flow fields . Here t he shock reflection gives t he highest speedup rates whereas t he wave interacti on exhibits t he lowest rates due to t he complex wave st ructures. Therefore we conclude t hat t he computational effort and t he memory requirements are prop ortional to t he number of cells in the adaptive grid. We emphas ize th at this can not be realized by Har ten 's original st rategy because t he resultin g scheme still involves t he full grid on t he finest level. This is based on the fact that locally expensive num erical fluxes are replaced by cheaper approximations, i.e., Har t en 's st rategy is based on a hyb rid flux evaluation where t he switching is perform ed by means of the multi scale analysis. However , th e adaptive strategy at hand reduc es the total number of flux evalua t ions not only th e number of expensive flux evaluat ions, i.e., we do not comput e t he chea p flux approximations in smoo t h regions at all. In combinat ion with the local multiscale t ransformation and its inverse, t he number of floatin g point opera t ions as well as the memory size of the resulting scheme is pr oportional to t he number of cells in t he ada ptive grid. Stability. The ob jecti ve of th e adaptive st rategy is to redu ce the computationa l complexity while maint ainin g t he accuracy in comparison to t he reference scheme on a uniform grid. In order to balance t he discretization error an d t he pert urbation error in (5.1) according to Corollar y 7 we need t he
130
7 Numerical Experiments
stability of the perturbation error in the sense of (5.3) . We would like to emphasize that the analytical results have only been verified for one-dimensional scalar problems, see Section 5.3. Here we consider a two-dimensional system. In addition, the numerical flux of the reference FVS does not match the assumptions of Theorem 5. Since for systems of conservation laws bounds on the global discretization error are not available we focus in the following just on the stability of the compression scheme . That is, instead of balancing discretization and perturbation error we fix the threshold value c and explore the dependence of the perturbation error on the level L, the number of time steps n and the temporal step size 7. For this purpose, the perturbation error is also listed in the Tables 7.2, 7.3, 7.4 as far as the computation could be performed on the available computers. Here the error eL is measured in the weighted II -metric where each conservative quantity is scaled by its global maximum in the computational domain, i.e.,
with et.; := (eL,k, ihEh ' Recall that the thresholding is also performed by means of the scaled details , see (3.38). We observe that the error is proportional to the threshold value c . For the shock reflection this error is even significantly smaller because the solution is piecewise constant. Hence no error is introduced by the truncation in regions where the solution is constant. To some extent, this also holds for the implosion. In case of the wave interaction the solution is no longer piecewise constant but only piecewise smooth exhibiting a genuine two-dimensional structure. We conclude that for all test cases the perturbation error does not increase with increasing number of refinement levels L. Note, that for an additional refinement level the number of time steps is doubled and the temporal step size is halved according to the CFL constraint. An additional error is introduced because the local flux evaluation is performed by (4.15) and not by (4.13) which is used in the analysis. Investigations in [CKMPOl] show that this does not affect the perturbation error provided the reference FVS is higher order accurate. This no longer holds for a first order FVS . In this case a significant error is observed. We conclude the discussion on the parameter studies with a remark on the "ideal" computation in the sense of (5.2) . Whenever the perturbation error is smaller than the discretization error, we waste performance in the sense of computational time and memory size, because the discretization error is still dominating. Conversely, we loose accuracy if the perturbation error is larger than the discretization error. Therefore the ideal computation of the parameter study corresponds to the refinement level where both the perturbation error and the discretization error are balanced. Finally, we would like to discuss the limitations of the adaptive scheme . For this purpose, we present some plots of the error, see Figures A.13, A.14 and A.15. We observe that the error accumulates near discontinuities. This
7.1 P ar amet er Studies
131
seems to be a cont ra dictio n because near discontinuities t he adapti ve scheme is expecte d to locally refine t he grid. However , erro rs introduced away from t he discontinuities are transported to a shock layer by means of th e cha racte ristics intersecting t here. On t he other hand , t he reference FVS is known to smear contact discontinuities. This leads to a smoot hing of t he solut ion which is detected by the multiscale analysis, i.e., near contact discontinuiti es t he adaptive scheme does not refine t he grid until t he finest level is reached. This confirms t hat the adapt ive scheme can, of course, not improve the soluti on of t he reference FVS but it can only provide a perturbation of this solut ion in a much mor e efficient way. The previous discussion indicates th at t he ada pt ive scheme meets t he design criteria, namely, (i) the computationa l effort and th e memory requirements are proportional to t he numb er of cells in t he ada ptive grid and (ii) t he perturbati on error is proporti onal to t he t hreshold value. Since in our par ameter st udies t he threshold value e is fixed , Le., e f: c(L), we can not dir ectly conclude on t he discretization err or . If we assume that t he discreti zation error for t he uniform grids of level L behaves like 2- a L we see from t he tables at which level the record ed perturbation err or is of compa ra ble size which gives an impression of the int err elation between t he overall accuracy and t he amou nt of computationa l work . However , t he computat ions on the full uniform grid of level L can only be realized for a small number of refinement levels L du e to t he huge amount of memory needed. 7.1.4 Hash Parameters
The efficiency of th e computations is significant ly influenced by the initi alization of t he hash maps MOe , GOe , L and v _map, d_map as well as adapgrid_map by which we store t he mask mat rices, t he sequences of local average s and significant det ails as well as t he ada ptive grid . For t his purpose, we carry out two parameter st udies concern ing th e parameters rF ill and nFaetor , see Sect ion 6.3, for the implosion configurat ion. Here one of t he two par ameters is fixed while t he ot her varies, see Table 7.5 and 7.6. We are primaril y int ereste d in t he imp act of th ese par ameters on t he computationa l time, the size of allocated memory and t he act ua l fill rates of t he above hash maps. Fir st of all, we investigate t he influence of t he par ameter nFaetor by which t he length of t he hash table is varied. The larger nFaetor is chosen the smaller becomes t he fill rate dFill of t he hash table, i.e., t he probability for a collision decreases, and visa versa. For this par ameter st udy rFill is fixed by 1, i.e., the size of th e allocated memory heap for sto ring t he keys and values is t hat of t he worst case corres ponding to the uniform finest grid. Considering Table 7.5 we not e t ha t t he memor y size increases with nFaetor because t he hash table becomes larger. At t he same time, t he act ua l fill ratio of the hash maps is reduced. However, choosing nFaetor larger t ha n 3 does not significant ly affect th e computationa l time. Conversely, if we redu ce nFaetor t hen t he size of th e hash t able becomes smaller which result s in
132
7 Nume rical Experim ent s Table 1 .5 . Implosion: par am et er st udy for nFactor wit h rFill
~
I
1
2
3
4
CPU [sec]
51
50
49
fill rat e [%] Memory [MB]
58 7.9
29 8.5
19 9.2
50 15 9.2
InFa ctor
=1
L=4
L= 5 CPU [sec] 249 238 233 235 fill rate [%] 40 20 13 10 Memory [MB] 23 25 26 31
L=6 CPU [sec] 1068 1044 1030 1032 fill rat e [%] 29 15 10 7 Mem ory [MB] 64 71 75 79
more collisions. T his leads to a significant increase of t he computationa l time because accessing data in t he has h maps consumes more time du e to collision ma nage ment. T he result s in Table 7.5 suggest to choose nFactor = 3 in view of an opt ima l computational time. T able 1 .6. Implosion: par am eter st udy for rFill wit h nFactor
L= 4 rF ill
0.12 0.25 0.5
1
2
49 77 8.0
49 19 9.3
49 10 12
0.08 0.17 0.35 0.7
1.4
CPU [sec] 50 161 fill rate [%] Memo ry [MB] 7.6
49 39 7.9
L =5 rFill
238 234 236 234 234 C PU [sec] fill rate [%] 166 78 38 19 10 Memory [MB] 19 18 19 22 31
L=6 rFill
0.04 0.08 0.15 0.3
0.6
1060 1035 1029 1036 1026 CPU [sec] 243 121 65 33 16 fill rat e [%] Memory [MB] 36 38 38 40 55
=3
7.2 Real World Appl icati on
133
Another parameter st udy was perform ed for rFill predicting t he relative size of the ada pt ive grid in comparison to N L of the uniform finest grid. This fact or influences the size of t he memory heap for sto ring the keys and values as well as the length of the hash t able. Considerin g Table 7.6 we not e t hat the size of allocate d memor y is t he larger t he higher the value for rFill. At t he same time, t he fill rate of t he hash t ables decreases, Le., the number of collisions redu ces. Thi s has a positive effect on the computationa l time. The actua l fill ra te s for the implosion problem can be deduced from Table 7.3 as 0.45, 0.28 and 0.14 for t he refinement levels L = 4, 5 and 6. Then we conclude from Table 7.6 th at t he memory size and the computationa l time are balan ced if we choose these numb ers for rFill. Since in general t hese numbers are not available before th e computation has been carried out, t he value rFill has to be predicted . Here t he par ameter st udy suggests t ha t the pr edict ed value should be at least as high as t he real value, i.e., overpredicting rFill is preferable to und erpredicting. If it is smaller t hen th e computational time increases significant ly du e to a higher collision rate. Finally, we would like to remark that t he collision rat e can be reduced if an opt imal prime numb er , i.e., a good choice for a modulo numb er, is chosen for t he length of t he hash t able. However , investi gations in the context of t he ada pt ive scheme show that the effort for determining an optimal number is much mor e expensive t ha n collisions caused by our choice for th e modulo number .
7.2 Real World Application Cha racterist ic for real world applications are m ultidim ensional configurations with complex geomet ries. These configurat ions can only be adequa tely discretized by mean s of unstructured or block- structured grids, respectiv ely. Although unstructured grids, e.g. arbitrary triangul ations, are easier to generate we prefer st ructure d grids, because th ey can be aligned with t he streamlines in t he flow field . In part icular , this is useful near surfaces in order t o resolve boundary layers. On th e other hand , many applications are steady state problems which are determin ed by t he limit of inst ati onar y computati ons. In t his conte xt implicit schemes are preferable to explicit schemes in order to use lar ger time ste ps when approaching t he steady state . In t he following we will verify that t he ada ptive concept can also be applied to an implicit FVS although no rigorous ana lysis can be prov ided so far. 7.2.1 Configurations
We consider two configurations which are relevant for the design of an airplane wing. Burnp . This configuration is considered in [RV81]. It is determined by a wall
134
7 Numerical Exp eriment s
wit h a bump where the bump is a circular arc with a secant of length l = 1 [m] an d a t hickness of h = 0.042 [m], see Figur e 7.4. We impo se a homo geneous flow field cha racte rized by th e free- stream quanti ties. Here we consider a t ra nsonic probl em cha rac terized by th e Mach numb er , th e temp erature and t he pressur e. Cha racteristic for t his configuration are t he kinks in th e sur face which influence t he solut ion in t he neighborhood of th ese singularit ies in t he geomet ry. In addition, a local supersonic domain develops caused by t he expa nsion along t he bump surface provided th e free-strea m velocity is sufficiently high . Fur ther downstream t he flow is compressed aga in . This lead s to a subsonic domain. According to physical properties t he subsonic and t he supersonic domain are sepa ra te d by a compression shock. Profile. This configurat ion is determined by a profile of an airfoil, see Figur e
Unit s
U oo
M a oo
------- ------
U oo
0.8
T oo
K
poo
Pa 101300
285
Fig. 7.4. Initi al configurat ion for circ ular arc bump
7.5. Here we choose the SFB401 cruise configuration accor ding to t he BAC 3-11/ RES/ 30/ 21 profile of Moir, see [Moi94, Pie97]. Cha racte ristic for t his problem is t he nonsymmetric shape of the profile and the two inflection points of t he lower surface where the surface cha nges from convex to concave and visa versa. Here we only consider axis-symmet ric free- stream condit ions, i.e., t he angle of attack 0: is put to zero. For sufficient ly high free- stream velocities two sup ersonic domains develop at the upp er and lower surface accompa nied by a local shock. In addition, a slip line occurs emanating from the trailing edge which is not a straight line du e to t he nonsymmetric shap e of t he profile. 7.2.2 Discretization The computations are performed by mean s of th e new flow solver QUADFLOW, see [BBMOO, BGMH+01]. Thi s solver merges t hree basic tools, namely, (i) an implicit FVS on unstructured meshes, (ii) th e local multiscale ana lysis and (iii) a grid generator. The reference implicit FVS is a high- ord er ENG scheme defined on uns tructured grids. The core ingredients of this scheme are (i) a genuinely multidimensional linear reconst ruction, (ii) t he limiter due to Venkat akrishnan [Ven95] and (iii) t he HLL Riemann solver
7.2 Real World Appli cat ion
---
Units
U oo
M a oo
135
U oo
0.76
Too
K
285
p oo
Pa
101300
Fig. 7.5. SFB401 cruis e configurat ion
modified according to [TSS94, BLG97]. For th e time discretization we apply local time st epping, i.e., for each cell we determine a local ste p size Tj ,k by a pr escribed CFL numb er CT according to
Hence the solut ion pro ceeds differently in tim e for each cell. This is only justified in case of st eady state problems. The implicit time discreti zation is resolved by a Newton- Krylov typ e method based on a matrix free GMRES and an ILU preconditioner. At th e boundaries we apply eit her slip conditions at t he solid wall, i.e., the norm al velocity component vanishes, or cha rac teristic boundary conditions for subsonic inflow and outflow boundaries in case of the bump configuration . For the profile we use a point vortex correct ion according to Thomas and Salas [TS86] at the far field boundary. The ada pt ive implicit FVS can be derived ana logously to the explicit case. However we have to replace th e explicit FVS (4.1) by an implicit FVS that can be writ ten in th e form
Here t he influence of the implicitn ess can be controlled by t he par ameter B E [0,1 ], e.g. B = 0 (explicit ), B = 1 (fully implicit) . For our computations we always use t he fully implicit scheme. Applying the local multiscale t ransform ation to these equations we derive t he local evolut ion equa tions for t he adaptive grid
in ana logy to (4.11). Since t he reference FVS is an unstructured solver , we can perform t he local flux evaluation by (4.16) instead of (4.15). Hence t he grid need not be locally uniform . In t his case it suffices to choose the gra ding par ameter q such t ha t t he local t ra nsformations are feasible. According to Corollary 4 t his is gua ra ntee d pro vided q 2: s = 1. For th e computations t he threshold value is put to E = 10- 3 . Moreover t he ada pt ive grid is updated
136
7 Numerical Ex periments
afte r N adap t ime steps inst ead of each time st ep in case of an instationary pr oblem. T he underlying grid genera to r is based on a block-structured concept where t he computational dom ain [l is decompo sed into severa l blocks, i.e., [l = Uk [lk· In Figur es 7.6 and 7.7 the decomp ositi on of the computational domain s for the above t est configurations are sketc hed.
Fig. 7.7. Cruise configura tion
Fig. 7 .6. Circular arc bump
In each block [lk we apply a st ructured grid genera t or. In view of a sparse repr esent ation of the nest ed grid hierar chy it is prohibited to st ore t he grid points. To t his end, the grid hierar chy is det ermined by a grid function which map s t he paramet er dom ain R onto the block [lk , see Section 3.8. In order to provide a fast computation of th e grid points as well as to cont rol grid properties, e.g. ort hogonality, smoot hness, etc., we repr esent the functi ons by B- splin es, i.e. , d
x(~) =
L Pk IT Nt (~i) , i
k
i =l
where Pk E Rd denot e t he cont rol points and N i4 t he B- splin e basis fun ct ions of ord er 4. In genera l, t he number of cont rol points is much smaller t han the number of grid poin t s corres ponding to t he finest resoluti on level L , see
[BMOO] . 7.2 .3 Discussion of Results
For each of t he two configurations we perform severa l computat ions with increasing number of refinement levels L and a fixed threshold value c. T he characterist ic valu es for t he und erlying discreti zati ons are summa rized in Tabl e 7.7. As initi al grid we choose t he uniform grid of levell , becau se the
7.2 Real World Application
137
Table 7.7. Discreti zation par amet ers
# bump profil e
blo cks N x ,o x N y ,o NT N adap CT 5x5 200 20 50 3 4
5x 5
250
25
50
initi al dat a are constant in the entire computat ional dom ain according to the free- stream conditions. After N a dap time st eps we adap t th e grid to t he current flow field by means of the local multiscale analysis. The st eady st ate is app roached afte r NT time st eps. In Figures A.16, A.17 and A.20, A.21 the local flow field nearby th e bump and t he profile, respect ively, as well as th e corresponding adapt ive grids are pr esent ed. Not e, th at due t o the sub sonic free- stream conditions the computational dom ain has t o be chosen sufficientl y larg e in ord er to avoid perturbations from th e far field boundary. For our computat ions we use 1.5 and 2.5 of th e bump span before and behind t he bump, respectively, and 2 bump spans in height . For th e profile we use about 20 wing spans in each dir ection. The plots of th e num erical results again verify t he reliability of th e multiscale analysis. All physic al meaningful effects have been detect ed. For both th e bump and the profile the shock is recognized. In addit ion, th e stagnation points of the bump configuration which coincide with t he kink s in th e sur face as well as the slip line in the wake of t he profile are det ect ed . Not e, t ha t in t he lat t er case there is no ju mp in the pressure field and t he normal velocity field . Since no singularity has been imp osed by t he initial data it is remarkable that t hese discontinuities evolve with pro ceeding t ime. Therefore it is import ant th at det ails on lower scales may pr edict significant det ails on higher scales within the pr ediction step , see Section 4.1.2 . Table 7.8. Paramet er study for the bump L
NL
4
19200 9.5
5
Nt;; e M S % [min]
IlrLill IlrLlloo
14 2E-9 3E-7
96800 2.5
14 2E-8 3E-7
6 387200 1.1
25 1E-8 5E-7
71548800 0.5
47 2E-8 2E-6
Again t he computationa l complexity behavior in case of st eady st at e problems is docum ent ed in Table 7.8 and 7.9. We note that t he number of cells in t he final adaptive grid is significantly smaller than that of the uniform grid on the finest resolution level L. In par ti cular , th e relative number Ng decreases
138
7 Nume rical Experiments Table 7.9 . Param et er st udy for the pro file L
NL
4
25600 28
N g eMS [rz.lh % [min]
5 102400 15 6 409600 8 71638400 4
Ilr£\1 00
30 6E-7 2E-3 51 3E-7 8E-5 92 lE-6 2E- 5 164 6E- 7 8E-6
much mor e rapidly with increasing numb er of refinement levels for the bump as in case of t he tes t configurations considered in t he pr evious section . This can be explained by t he local st ruct ure of t he singularities. For the profile t he relat ive number N9 does not decrease so rapidly with increasin g refinement level because t he slip line t raverses th e ent ire wake whereas the singularities occur ring in the bump configuration are local. In contrast to inst ati onar y problems, we have to consider t he asy mpt ot ic behavior of the solut ion which is expected to approach a steady state . An indicator is defined by t he difference rj,k := vj,t l - vj,k of the conservati ve qu anti ties on two successive time levels which frequently referr ed to as the (temporal) residual. In Table 7.8 t he maximal residualllrLlloo and the average residual llrLill defined by
IlrLlloo
:= . max
IlrLill
:= . max
. max
t=l, ... ,m (J,k )E9 L .€
'l.=l, ... ,m
L
lrJ ,k,il (I VJ,kl/lnl)l rj,k,il
(j, k) E9 L ,€
are listed . For our computations t he volume of t he computational domain is about Inl = 10 (bump) and Inl = 1271 (profile) . It t urns out that the maxim al and average residu al are redu ced by four ord ers of magnitude. during t he computations for both configurations. Thi s is verified by t he plots of t he tempora l variation of t he maxim al residu al for t he density, see Figur e A.18 and A.22. We observe tha t between two adaptation ste ps the residua l decreases almost monotonically. However afte r each grid ada ptation th e residual significantly increases by severa l ord ers of magnitude. This is caused by t he t hresholding t hat is performed wit hin t he local multiscale ana lysis. Note , t hat t he new residu al after adapting t he grid is in t he order of the t hreshold value c. Nevert heless the residu al decreases more st rongly after each ada ptation ste p . Approaching th e steady state in not only reflected in t he decrease of the residu als, but also in t he ra tio of the numb er of grid points before and after t he adaptation step . In Figur e A.19 and A.23 this ratio is presented . Here aga in we observe an asymp totic behavior. In the limit we approac h 1, i.e., t he number of grid points is maintained.
A Plots of Numerical Experiments
Fig. A.I. Shock reflect ion: density cont our s (L
= 8, T = 0 [s])
Fig. A.2 . Shock reflecti on : ada ptive grid (L = 8, T = 0 [s])
140
A Pl ot s of Nu merica l Ex periments
Fig. A.3. Shock reflectio n: densit y contour s (L
Fig. A.4. Sho ck reflection : adaptive grid (L
= 8,
= 8,
T
T
= 3.2 X 10- 4
= 3.2 X
[s])
10- 4 [s])
A Plots of Numerical Experiments
Fig. A .5 . Implosion : densit y contours (L = 9, T = 0 [s])
Fig. A.6. Implosion : adaptive grid (L
= 9, T = 0 [s])
141
142
A Plots of Numerical Experiments
Fig. A.7. Implosion: density contours (L
Fig. A.B. Implosion: adaptive grid (L
= 9,
T
= 3.6125 X
10- 4 [s])
= 9, T = 3.6125 X 10- 4
[s])
A Plots of Numerical Exp erim ents
Fig. A.9 . Wave int eracti on: density conto urs (L
Fi g. A .lO. Wave inte rac t ion: ada pt ive grid (L
= 6,
T
= 0 [s])
= 6, T = 0 [s])
143
144
A Plots of Numerical Exp erim ents
F ig . A. l1. Wav e int eracti on : density cont ours (L = 6, T = 0.15 [s])
Fig. A .12. Wave inter acti on : ada ptive grid (L
= 6, T = 0.15 [s])
A Plots of Numerical Exp erim ents
Fig. A .13 . Shoc k reflection: conto urs of erro r in density (L
= 7, T = 3.2 X 10- 4
Fig. A.14. Im plosion: contours of error in density (L = 7, T = 3.6125
X
145
[s])
10- 4 [s])
146
A Pl ot s of Numerical Experiment s
Fig. A .I5 . Wave int eraction: conto urs of error in density (L
= 5, T = 0.15 [s])
A Pl ot s of Numerical Experiment s
Fig. A.16. Bump: Mach cont ours (L
-
I
-
Fig. A .17. Bump: adaptive grid (L
= 7)
= 7)
147
148
A P lots of Numerical Experiments
0 .1 0 .01
..t+ +
++ + ++ + ++
++
0.001
+
,t++*+
+
0 .0001
+ +
+++
+:
++
+
+
trip- \.+
1e-05
t+ ~.t
~
~
~ 1j..
+++
V~+~ .;+;.t ......
1e-06 1e-07
.... +
++
o
80
40
120
160
200
n Fig. A.IS . Bump: temporal variation of maximal density residu al (L
4 0
3 .5
-..-l
.w
ro
H
3
.w 2.5 s:: 0 2
•..-l
0..
ro 1.5 H o 1
•..-l
+
0 .5 0
40
80
120
160
200
n Fig. A.19 . Bum p: te mporal variation of grid size rati o before/after ada ptation (L = 7)
= 7)
A Plots of Numerical Experiments
Fig. A.20. Profile : Mach contours (L = 7)
Fig. A .21. Profile : adaptive grid (L
= 7)
149
150
A Plots of Numerical Experim ents
0.1 0 .01 r-I
cO
0.001
;::::l
+
+ +
\
+
+
'D
•.-1
Ul Q)
p::
0.0001 1e-05 1e-06
o
50
100
150
200
250
n Fig. A.22 . Profil e: t emporal vari ation of maximal density residu al (L
0
4
•.-1
4-l
cO H
3
4-l
I=:
-.-I
0
o,
2
ro
•.-1
H
1
(9
0 0
50
100
150
200
250
n Fig. A.23 . Profile: t emporal vari ation of grid size ratio before/after ada ptat ion (L = 7)
= 7)
B The Context of Biorthogonal Wavelets
This section provides a short introduction to (biorthogonal) wavelet t heory. Here we focus on some basic aspects that are restricted to our purposes. For a general overview on wavelet theory we recomm end st andard t extbooks such as t he recent book by A. Cohen [CohOO] . We start with a genera l multiresolution analysis following th e methodology of W . Dahmen [Dah94, Dah95 , Dah96] and collaborators [CDP96]. In particular, we out line the connection between stability and biorthogonal wavelets. As an example for th e general setting we consider wavelets on t he real line according to [CDF92].
B .l General Setting In t he sequel, we consider a Hilbert space ! H = 1-l(rt) of functions endowed with an inner product (-, -)t{ and associat ed norm II . lit{ := (-, . ) ~2 . Then a sequence S = {Sj LENa of closed linear subspaces Sj C 1-l is called a m ult iresolution or multiscale sequence, if the subspaces are n est ed, Le., So C Sl C .. . C
s, C Sj+l
C .. . 1-l
and S is dense in 1-l, i.e., clos-,
(U .
l ENa
Sj) = 1-l .
The nest edness of the multiresolution sequence implies t he existe nce of the compl em ent or wavelet spaces W j such t ha t
1
Not e, that st ability is best investigated in £ 2 . Similar results ca n not b e derived in £1. However , in Sect . 2, t he box wavelets and their modification hav e to b e scaled with respect t o £1.
152
B The Context of Biorthogonal Wavelets
B.l.l Multiscale Basis
We now assume that the subspaces Sj and W j are spanned by sets of basis functions Pj := {cpj,k; k E I j},
enumerated by the index sets I j and Jj, i.e., Sj = clos-, (span{Pj}) = : S(Pj),
W j = clos-, (span{tlij}) =: S(tli j).
We refer to Pj as the basis of scaling junctions? and tlij as the wavelet basis. Here the index sets are either finite or at most countable sets . In particular, in the finite case we have N j H = N, + K j with N j := #Ij , K j := #Jj. For later use it is convenient to interpret a set of functions G j , e.g., G j = Pj or Gj = tlij, as a column vector with respect to a fixed but unspecified ordering of basis functions . Then it is possible to represent any function U E span{G j} by U
=
GJ u := L
Uk OJ,k,
k
The collection {Gj}jENo is called uniformly stable, if there exist constants c, C > 0 independent of j such that the Riesz property
(B.l) holds for all u E R#8; (see [CDP96]). In particular, if Gj is an orthonormal system, then the Riesz property holds with" = " instead of ":s" and c = C = 1. The Riesz property (B.l) with uniform constants ensures the uniqueness of the expansion of U with respect to the basis Gj . If the basis G j satisfies the Riesz property then it is called a Riesz basis for the space S (G j) . Keeping our application in mind , we confine the discussion to uniformly stable Riesz bases, Le., {Pj}jEN o and {tlij}jENo satisfy the Riesz property (B.l) . In the context of principal shift-invariant spaces Sj, i. e., if 1i is a Hilbert space over Rd and S satisfies
f
E
s,
if and only if
the notion of a multiscale sequence coincides with the concept of Mallat's multiresolution analysis [Ma189]. An example is discussed in Sect. B.2. From the nestedness of S and the uniform stability of the Riesz bases, we conclude the existence of bounded linear operators Mj,o and Mj,l such that the twoscale relations (B.2) or componentwise 2
Here the notion of scaling function is used in a generalized sense .
B.1 General Set ting . t. do
-
M l ,o
Ul
Uo
dl
Ml,l
.>
-
M L- l ,o U L-l
.>
d2
UL
.>
d L- l
M L-l , l
Fig. B.l. P yr ami d scheme of the mul t iscale tran sformati on
UL- l
-
d L- l
G L- 2 , l
GL-l,O
UL
0, t here is a du al funct ion 0 E L 2 (R) = 1,fir0 E L 2 (R) which is refinab le, has compact support [1 (
