Gerald Warnecke (Ed.) Analysis and Numerics for Conservation Laws
Gerald Warnecke Editor
Analysis and Numerics for Conservation Laws With 236 Figures and 18 Tables
123
Editor
Gerald Warnecke Institut für Analysis und Numerik Otto-von-Guericke-Universität Magdeburg Postfach 4120 39016 Magdeburg, Germany e-mail:
[email protected] Library of Congress Control Number: 2005922932
Mathematics Subject Classification (2000): 35L65, 65M99, 65Z05, 76N15
ISBN-10 3 -540-24834-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-24834-7 Springer Berlin Heidelberg New York This work is subject to copyright. 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 thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered 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 therefore free for general use. Typesetting by the author using a Springer TEX macro package Cover design: Erich Kircher, Heidelberg, Germany Printed on acid-free paper
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Preface
What do a supernova explosion in outer space, flow around an airfoil and knocking in combustion engines have in common? The physical and chemical mechanisms as well as the sizes of these processes are quite different. So are the motivations for studying them scientifically. The supernova is a thermo-nuclear explosion on a scale of 108 cm. Astrophysicists try to understand them in order to get insight into fundamental properties of the universe. In flows around airfoils of commercial airliners at the scale of 103 cm shock waves occur that influence the stability of the wings as well as fuel consumption in flight. This requires appropriate design of the shape and structure of airfoils by engineers. Knocking occurs in combustion, a chemical process, and must be avoided since it damages motors. The scale is 101 cm and these processes must be optimized for efficiency and environmental considerations. The common thread is that the underlying fluid flows may at a certain scale of observation be described by basically the same type of hyperbolic systems of partial differential equations in divergence form, called conservation laws. Astrophysicists, engineers and mathematicians share a common interest in scientific progress on theory for these equations and the development of computational methods for solutions of the equations. Due to their wide applicability in modeling of continua, partial differential equations are a major field of research in mathematics. A substantial portion of mathematical research is related to the analysis and numerical approximation of solutions to such equations. Hyperbolic conservation laws in two or more space dimensions still pose one of the main challenges to modern mathematics. This is due to the fact that the fundamental question of an existence theory for solutions to these equations has remained an open problem for many decades, despite intensive efforts of some of the leading mathematicians during the last fifty years. Due to their importance in applications the development of efficient numerical methods had to proceed despite the fact that mathematics could not prove that the solutions, to be approximated computationally, actually exist. The preferred practice is to seek numerically objects that are mathematically proven to exist and whose properties are well understood.
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Preface
In this situation it is important that analysis and numerical computations, including the work in fields of application, are studied in close cooperation. By numerical experiments we can explore the structure and properties of solutions in the hope of inspiring the search for the right analytical framework for these equations. Researchers developing computational methods or applying numerical method need as much analytical information on solutions as possible in order to gain confidence in their computations. Especially applied researchers have to be aware of this situation in order to make the right assessments in their comparisons with experimental or observational data. Hyperbolic conservation laws for fluid flows in conjunction with other mechanisms such as material elasticity, magneto-hydrodynamics or combustion were at the center of a major research effort in Germany in recent years. The priority research program Analysis and Numerics for Conservation Laws was funded by the German research foundation Deutsche Forschungsgemeinschaft (DFG) for a period of six years from 1997. The program consisted of three periods of two year grants for about 25 projects in each period including funds for visitor programs and workshops. The participating groups came from astrophysics, fluid mechanics, mathematical analysis and numerical mathematics. The span of research interests was from specific applications to fundamental mathematical questions. A number of workshops were organized in each year and in 2000 the 8th International Conference on Hyperbolic Problems: Theory, Numerics and Applications was held in Magdeburg, Germany. Another special event was a Symposium on Entropy jointly organized with two other DFG priority research programs Interacting Stochastic Systems of High Complexity as well as Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems (DANSE). Together with the two co-organizers of the symposium Andreas Greven and Gerhard Keller specially selected contributions to the topic were compiled in a book titled Entropy. The diversity of topics, represented in the volume you are looking at, was one of the strengths of the research program. It brought together research groups of very different background most of which were interacting with each other for the first time. This volume can be seen as a research report of the priority research program. It contains contributions from most participating projects and gives an overview of some results achieved in the program. The authors intend to convey their results to readers outside of their own particular field. They cover a wide range of conservation laws modeling, for instance, bubbly flows, retrograde and BZT fluids, detonation waves in combustion, magnetoplasmadynamic propulsion of space craft, solar physics, chemotaxis in mathematical biology, and type Ia supernovae. Kinetic models using the Boltzmann equation or the Boltzmann-Peierls equation are explored in relation to conservation laws. The numerical approaches range over finite volume methods, central schemes, the method of transport, discontinuous Galerkin methods, kinetic schemes to meshless particle methods. The efficient use of level-set methods for problems needing better interface resolution was a research topic for a number of groups. The majority of the contributions deal
Preface
VII
with the development and study of properties of numerical schemes. Additionally, mathematicians present results on error estimates for scalar conservations laws, existence of traveling wave solutions, structural stability, relaxation dynamics and scaling limits, as well as the value of dual a posteriori error estimates for adaptive algorithms. For some computational results a color scale is a convenient way to plot field variables. Therefore, a number of color plates have been included in the back. I would like to take this opportunity to thank those who helped in many ways. Robert Paul K¨ onigs and Bernhard Nunner were in turn the program officers of the DFG for mathematics during the conception and implementation of the program. They were exceptionally helpful with very sound advice and contributed substantially to the success of this program. A big thanks also to all participating colleagues for their input into devising the program and their continual support in running the program, this especially to the members of the coordinating board Dieter H¨ anel, Wolfgang Hillebrandt, Rupert Klein, Dietmar Kr¨ oner and Willi J¨ ager. Other colleagues supported this effort by taking on the difficult task of refereeing the project proposals that always outnumbered the budgetary possibilities of the DFG. They are Wolfgang Hackbusch, Rolf Jeltsch, Wilhelm Kegel, Wilhelm Kordulla, Stephan Luckhaus, Hans Ruder, Martin Sommerfeld, Peter Szmolyan and Harry Yserentant. Most important to me was the success of a quite substantial number of doctoral students funded within the program or taking part in activities generated by the program. I enjoyed their lively interaction very much and would like to thank them for their contributions to the activities and the scientific results of this program. The manuscripts of this book were handled by my secretary Stephanie Wernicke and the final draft for the publisher was expertly compiled by R¨ udiger M¨ uller. Magdeburg December 2004
Gerald Warnecke
Contents
Wave Processes at Interfaces S. Andreae, J. Ballmann, S. M¨ uller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Numerics for Magnetoplasmadynamic Propulsion J. Heiermann, M. Auweter-Kurtz, C. Sleziona . . . . . . . . . . . . . . . . . . . . . . . 27 Hexagonal Kinetic Models and the Numerical Simulation of Kinetic Boundary Layers H. Babovsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 High-resolution Simulation of Detonations with Detailed Chemistry R. Deiterding, G. Bader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Numerical Linear Stability Analysis for Compressible Fluids A.S. Bormann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Simulation of Solar Radiative Magneto-Convection M. Sch¨ ussler, J.H.M.J. Bruls, A. V¨ ogler, P. Vollm¨ oller . . . . . . . . . . . . . . . 107 Riemann Problem for the Euler Equation with Non-Convex Equation of State including Phase Transitions W. Dahmen, S. M¨ uller, A. Voß . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Radiation Magnetohydrodynamics: Analysis for Model Problems and Efficient 3d-Simulations for the Full System A. Dedner, D. Kr¨ oner, C. Rohde, M. Wesenberg . . . . . . . . . . . . . . . . . . . . . 163 Kinetic Schemes for Selected Initial and Boundary Value Problems W. Dreyer, M. Herrmann, M. Kunik, S. Qamar . . . . . . . . . . . . . . . . . . . . . 203
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A Local Level-Set Method under Involvement of Topological Aspects F. V¨ olker, R. Vilsmeier, D. H¨ anel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Hyperbolic Systems and Transport Equations in Mathematical Biology T. Hillen, K.P. Hadeler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Travelling Waves in Systems of Hyperbolic Balance Laws J. H¨ arterich, S. Liebscher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 The Role of the Jacobian in the Adaptive Discontinuous Galerkin Method for the Compressible Euler Equations R. Hartmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 The Multi-Scale Dust Formation in Substellar Atmospheres Ch. Helling, R. Klein, E. Sedlmayr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Meshless Methods for Conservation Laws D. Hietel, M. Junk, J. Kuhnert, S. Tiwari . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Simulations of Turbulent Thermonuclear Burning in Type Ia Supernovae W. Hillebrandt, M. Reinecke, W. Schmidt, F.K. R¨ opke, C. Travaglio, J.C. Niemeyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Hyperbolic GLM Scheme for Elliptic Constraints in Computational Electromagnetics and MHD Y.J. Lee, R. Schneider, C.-D. Munz, F. Kemm . . . . . . . . . . . . . . . . . . . . . . 385 Flexible Flame Structure Modelling in a Flame Front Tracking Scheme H. Schmidt, R. Klein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Riemann-Solver Free Schemes T. Kr¨ oger, S. Noelle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Relaxation Dynamics, Scaling Limits and Convergence of Relaxation Schemes H. Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Multidimensional Adaptive Staggered Grids S. Noelle, W. Rosenbaum, M. Rumpf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 On Hyperbolic Relaxation Problems W.-A. Yong, W. J¨ ager . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Appendix: Color Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
Wave Processes at Interfaces Sigrid Andreae1 , Josef Ballmann1 , and Siegfried M¨ uller2 1
2
Lehr- und Forschungsgebiet f¨ ur Mechanik der RWTH Aachen sigrid.andreae|
[email protected] Institut f¨ ur Geometrie und Praktische Mathematik, RWTH Aachen
[email protected] Summary. We investigate the interaction of shock waves in a heavy gas with embedded light gas bubbles next to a rigid wall. This may give insight regarding cavitation processes in water. Due to the highly dynamical, unsteady processes under consideration we use an adaptive FV scheme for the computations to resolve accurately all physically relevant effects. The results are validated by comparison with tube experiments.
1 Introduction The formation and collapse of vapor bubbles in a liquid is called cavitation. Lord Rayleigh discovered that pressure waves emitted during the process of cavitation [Ray17] may damage solids, e.g., marine screw propellers. Since then, the mechanism of cavitation damaging has been subject of experimental [Lau76, LH85] and analytical research. However, it is still unclear whether the shock and rarefaction waves or the liquid jet onto the solid is the main reason for the erosion of the material. The loading on an elastic-plastic solid exposed to shock and rarefaction waves in water was investigated by Specht in [ASB00]. Hanke and Ballmann showed one-dimensional results for a bubble collapse in water in [HB98]. Cavitation is induced by a pressure drop in the liquid below vapor pressure. Such a pressure decrease may occur due to local acceleration of the liquid flow caused by geometrical constraints, e.g., if the liquid flows through a narrow orifice or around an obstacle. In case the pressure drops below vapor pressure, the liquid bursts and creates a free surface filled with gas and vapor – the bubble. Due to changes in the flow field, the pressure in the liquid may increase afterwards causing the bubble to collapse. The collapse is accompanied by strong shock and rarefaction waves running into the bubble and the surrounding liquid. The shock wave focuses in the center of the bubble. This leads to extreme physical states in the interior. In addition, the shrinking of the bubble leads to a compression of the vapor. Both effects evoke an increase
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S. Andreae, J. Ballmann, S. M¨ uller
of pressure which bulges the bubble. Hereby, a dynamic oscillation process is initiated which finally leads to the collapse of the bubble. If the collapse takes place next to a solid, the pressure distribution becomes asymmetric and a liquid jet develops [PL98] which is either directed towards or away from the solid boundary. The direction of the jet depends on the elasticity of the solid and on a ratio determined by the initial distance between the boundary and the center of the bubble divided by the maximum extension of the bubble. This has been shown experimentally by Brujan et al. in [BN+ 01], [BN+ 01a]. In order to investigate experimentally the dynamics of a bubble collapse, the bubble is produced by a laser pulse. Thereby, the fluid is heated in the focus of the laser and forms a small, hot gas bubble at very high temperature. This experimental setup provides an exact positioning of the bubble. The processes taking place in the interior of the collapsing and oscillating bubble and the prediction of onset and extent of the cavitation damaging are still subject of theoretical and experimental research. However, small time and space scales as well as the complicated dynamics make an experimental approach difficult. Therefore numerical investigations are developed to reveal information about the wave dynamics in the fluid as well as the damaging of the solid. Of particular interest are pressure contours and velocity vectors in the liquid phase as requested in [BN+ 01a]. The primary objective of the present work is to provide an accurate prediction of all occurring wave phenomena. This concerns wave interactions among each other, with phase boundaries or neighboring solids. Of particular interest is the occurrence of instabilities as, e.g., the Richtmyer-Meshkov instability. Since all present methods for simulating two-phase flows suffer from pressure oscillations at the phase boundary, we use a very dense and heavy gas instead of water. The occurring wave phenomena are expected to be qualitatively comparable with those in water. For two-phase flow problems with different equations of state the phase boundary can be tracked or treated in a Lagrangian manner as a sharp, interior boundary using two meshes [Dick96]. The latter suffers from the drawback that the mesh has to be updated in every step which is expensive and time consuming and may result in a poor mesh quality for large displacements of the interface. Instead, we track the phase boundary using a level set method. Consequently, the phase boundary is represented as a mathematically sharp boundary as we will explain in Section 2. We do not implicate surface tension and mixing of the two fluids. For the sake of completeness, it is to be mentioned that for a two phase flow of one fluid a homogenized approach is possible, as will be presented by Voß in [Vos04]. There, one equation of state is used for the liquid as well as the gaseous phase and the so-called mixture region. Here, even states consisting of gas and vapor fractions can be modeled. The small time scales of the unsteady problem require the numerical scheme to be highly efficient regarding computational time and memory requirements. This is realized by a local grid refinement strategy. Furthermore,
Wave Processes at Interfaces
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all physically relevant phenomena have to be reliably detected and adequately resolved. Moreover, the scheme has to be robust and must not exhibit numerical oscillations, e.g., pressure oscillations at the phase boundary. The details of the numerical scheme are presented in Section 3. In Section 4 numerical results for a bubble collapse near a rigid wall are presented and the arising dynamic wave pattern is discussed.
2 Level Set For modeling two-phase flows, there are mainly two different approaches to treat the two media, a fitting of the phase boundary with two separate grids connected by interface conditions, [Dick96], or one grid with a suitable algorithm to track the phase boundary. Since in the first case the grid has to be redesigned in every time step which is very time consuming, we use only one grid and the level set method to distinguish the two fluids. The level set method, proposed by Osher and Sethian in [OS88], is a tool to track propagating interfaces without an explicit description like a function of the interface under consideration. Instead a scalar field given in the domain is used to represent the motion of the interface. Consider the case of two domains Ω1 , Ω2 separated by a contact surface Γ . Now, a scalar field φ = φ(x, t) is introduced which is φ(x, t) < 0 for Ω1 x ∈ Ω1 and φ(x, t) > 0 for x ∈ Ω2 , see Fig. 1. The Γ Ω2 interface Γ is evolved in time by the fluid velocity v. Therefore, we may describe the time evolution of Fig. 1. Domains. the scalar field φ by ∂φ + v · ∇φ = 0 . (1) ∂t There are two different methods to exploit the evolution of the scalar field for tracking a moving interface. The most common approach suggested by Osher and Sethian in [Set96] is to define the scalar field as a smooth, signed distance function to the front under consideration, whereby the material interface corresponds to φ = 0. Sussman et al. used this approach in computing incompressible two phase flows in [SSO94, SA+ 99, SF99]. The smoothness of φ has to be sustained by a reinitialization after each time step. This way, it is guaranteed that the level set itself will not steepen and develop shocks. A disadvantage of this method is the loss of conservativity. Nguyen et al. suggested methods to recover the conservativity, see [NGF02]. Here, we follow an idea of Mulder et al. [MO92] where the level set function is not a smooth but a discontinuous scalar field. Initially, we assign φ(x, t) = −1 for x ∈ Ω1 and φ(x, t) = +1 for x ∈ Ω2 . This notation for φ is sometimes called “color”-function. The sign (color) of φ decides which fluid occupies which domain. Thereby, the choice of the equation of state is
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S. Andreae, J. Ballmann, S. M¨ uller
controlled. It has to be noticed that we track the jump in φ and not the zero level set. Multiplying equation (1) with the density and employing the continuity equation of fluid dynamics yields a conservative form of the transport equation for φ, i.e., ∂φ + ∇ · (φv) = 0 , (2) ∂t Written in this form equation (2) can be added to the system of conservation equations (1) as an additional equation. The main advantage of this approach is the preservation of conservativity.
3 Governing Equations and Method of Solution The fluid flow is modeled by the time-dependent 2D Euler equations for compressible fluids. Appending the evolution equation (2) this leads to the system of conservation equations ∂ U dV + F · n dS = 0 with (3) ∂t V ∂V ⎛
⎞ ⎜ v ⎟ ⎟ U=⎜ ⎝E ⎠, φ
⎛
⎞ v ⎜ v ◦ v + p1 ⎟ ⎟ F=⎜ ⎝ v ( E + p) ⎠ . φv
Here, U is the array of the mean conserved quantities: density of mass, momentum, specific total energy and level set. p is the pressure and v the fluid velocity. The quantity V denotes a time-independent control volume with the boundary ∂V and the outer normal n. The flux F contains only the convective terms. Since the two fluids under consideration are gaseous both, there is no need to deal with the surface tension at their contact surface. The system of equations is closed by the perfect gas equations of state for the fluid i = 1, 2 present in the domain Ωi , i.e. the affiliated thermal equation, p = Ri T , and the caloric equation, e = cvi T . Herein, e is the internal energy and T the temperature. cvi and Ri are the heat capacity at constant volume and the special gas-constant for fluid i = 1, 2, respectively. The material properties, cvi and Ri , are listed in Table 1. The evaluation of the equations of state is governed by the scalar field φ, i.e.,
p1 = p (R1 , , T ) : φ < 0 p= . (4) p2 = p (R2 , , T ) : φ > 0 The conservation equations (3) are discretized by a finite volume method. The convective fluxes are determined by solving quasi–one dimensional Riemann problems at the cell interfaces. For this purpose we employ a two-phase
Wave Processes at Interfaces
5
Roe Riemann solver designed for the coupled system of the 2D Euler equations and the evolution equation (2) of the level set φ. For the construction of this solver we proceed similarly to [LV89] for real gases. In order to avoid non-physical expansion shocks we use Harten’s entropy fix. The spatial accuracy is improved by applying a quasi one-dimensional second order ENO reconstruction. Due to the strong dynamic behavior of the considered flow problems the time integration is performed explicitly. In order to properly resolve all physical relevant phenomena we need a very fine discretization of the computational domain. Due to the heterogeneity of the flow field, this high resolution is not needed throughout the entire computational domain but only locally near discontinuities. For this purpose we employ a dynamic local grid adaptation strategy to resolve the physically relevant phenomena at the expense of possibly few degrees of freedom and correspondingly reduced storage demands. The main distinction from previous work in this regard lies in the fact that we employ here recent multi-resolution techniques, see [M¨ ul02]. The starting point is to transform the arrays of cell averages associated with any given finite volume discretization into a different format that reveals insight into the local behavior of the solution. The cell averages on a given highest level of resolution are represented as cell averages on some coarse level where the fine-scale information is encoded in arrays of detail coefficients of ascending resolution. This requires a hierarchy of meshes. The multiscale representation is used to create locally refined meshes. For details we refer to [M¨ ul02]. Following Mulder [MO92] we chose φ as a color function in our computations. Mulder observed in [MO92] that using this formulation of φ the pressure shows spurious oscillations at the phase boundary. To reduce these oscillations we use averaged pressure and energy equations near the interface, i.e., ⎫ ⎧ : φ < − ⎬ p1 ⎨ , (5) p = (1 − α (φ)) p1 + α (φ) p2 : |φ| < ⎭ ⎩ : φ > p2 ⎧ ⎫ cv1 : φ < − ⎬ ⎨ (1 − α (φ)) cv1 + α (φ) cv2 : |φ| < e = T cv = T . (6) ⎩ ⎭ : φ > cv2 Here, the function α (φ) is chosen as a linear interpolation between 0 and 1 in the interval [−, ], i.e., α (φ) = (φ/ + 1) /2 for |φ| < . For our computations we chose = 0.5. Results of a comparison between the approximate Riemann solver using this averaging method and an exact Riemann solver are given in Figs. 2(a) and 2(b). Since φ is initialized by +1 for fluid 1 and by −1 for fluid 2, has to be chosen less than 1 to make sure that the modification of the pressure law is only applied in the vicinity of the interface. Note that the initial jump of φ is smeared by the FV scheme. The width of this numerical transition layer depends on the underlying grid resolution. It becomes smaller with finer
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S. Andreae, J. Ballmann, S. M¨ uller
grids. In particular, when the material boundary is a phase boundary and different equations of state have to be applied on both sides, a grid adaption strategy is strongly recommended to provide a high resolution of the interface. From this point of view, equation (5) describes not really a physically relevant phase transition, it serves more as a numerical stabilizer of the phase boundary.
4 Numerical Results The current work focuses on the ability of the scheme to accurately resolve the dynamics and wave pattern occurring in the presented test configurations for two-fluid flow. At first, we validate our scheme using experiments performed by Haas and Sturtevant, [HS87]. Herein, a shock runs across a bubble filled with helium in the one case and with R22 gas in the other. R22 is the heavy refrigerant chlorodifluromethane (CHClF2 ). In Table 1 the physical properties of the gases under consideration are given. The surrounding fluid is air in both cases. We compare our numerical results with the Schlieren photographs taken by Haas and Sturtevant. The third configuration is a helium bubble surrounded by R22 and placed next to a rigid wall, with initial conditions corresponding to an explosion problem. The main focus lies on the interaction of the emitted waves with the wall. Since we expect the occurring wave phenomena have something in common with the formation and collapse of a cavitation bubble, this configuration indicates possible causes for the damaging mechanisms accompanying cavitation in the related experiments. The Table 1. Molecular weight umol , special gas-constant R, ratio of specific heats γ and speed of sound c (at 293.15 K, 101, 35 kPa) for air, helium and R22. fluid
umol [103 kg/mol]
R [J/kg/K]
γ
c [m/s]
air helium R22
28.964 4.003 864.687
287.0 2077.0 96.138
1.4 1.66 1.178
343.3 1007.4 184.0
characteristic physical quantity in dealing with wave interactions with boundaries is the acoustic impedance c. Herein, c is the speed of sound. The ratio of the acoustic impedances of two fluids governs what happens to a shock wave traveling through fluid 1 and impinging on the phase boundary between fluid 1 and 2. According to the acoustic wave theory, the impinging shock wave is split up in a transmitted part traveling through Fluid 2 and a reflected part. The larger the jump of the acoustic impedance the more energy is reflected. In case, (c)1 >> (c)2 most of the energy is reflected with a phase change of 180◦ . If (c)1 0 : U ∈ C([0, T ], H s ) .
(19)
s
Namely, [0, T ) is the maximal time interval of H existence. Note that T depends on G and may tend to zero as goes to a certain singular point, say 0. In order to show that lim→0 T > 0, which means the stability (see [KM82, Ma84]), we make the following convergence assumption: there exists T∗ > 0 and U ∈ L∞ ([0, T∗ ], H s ) for each , satisfying 8" $ U (x, t) ⊂⊂ G, x,t,
such that for t ∈ [0, min{T∗ , T }), sup |U (x, t) − U (x, t)| = o(1), x,t
sup U (·, t) − U (·, t)s = O(1) t
as tends to the singular point. With such a convergence assumption, we are in a position to state the following fact established in [Yo01]. ¯ (x, ) ∈ G0 ⊂⊂ G for all (x, ), U ¯ (·, ) ∈ H s with Lemma 4.2 Suppose U an integer s > d/2 + 1, and the convergence assumption holds. Let [0, T ) be the maximal time interval such that (18) has a unique H s -solution U ∈ C([0, T ), H s ). Then T > T∗ for all in a neighborhood of the singular point. Thanks to Lemma 4.2, our task is reduced to find a U (x, t) such that the convergence assumption holds. Below, we will use this lemma with G replaced by its compact subsets. It is remarkable that similar lemmas can be formulated for other evolution equations. For IVPs of (2) satisfying the above stability condition, such a U (x, t) was constructed in [Yo92, Yo99a] with the classical matched expansion method. The construction additionally needs the following two technical assumptions.
On Hyperbolic Relaxation Problems
509
(a) the limiting inner problem (x is a parameter here) dI˜ ˜ = Q(I) dτ
˜ 0) = U ¯ (x, 0) with I(x,
has a unique solution I˜ ∈ C([0, +∞), H s+3 ), with s > d/2 + 1 an integer, which takes values in a convex compact subset of G and decays exponen˜ +∞) in H s+3 (Ω) as τ goes to infinity; tially to a function I(x, (b) the equilibrium manifold can be expressed as $ " E = U = E(u) : u ∈ U ⊂ Rn−r , where U is open and E is a smooth diffeomorphism from U to E. Note that (a) relies on the non-oscillation assumption, while (b) is convenient and for many examples also appropriate. With the above technical assumptions, we seek formal asymptotic approximations, to the initial-layer solution U of (2), of the form Um (x, t)
=
m
k
Uk (x, t) +
k=0
m
k Ik (x, t/)
(20)
k=0
with 1 ≤ m ≤ s − 1 − [d/2]. By our construction, the leading term I0 (x, t/) of the initial-layer correction (the second sum) is ˜ t/) − I(x, ˜ +∞) I0 (x, t/) = I(x,
(21)
and the leading term U0 of the outer expansion (the first sum) satisfies the initial condition ˜ +∞). U0 (x, 0) = I(x, Moreover, U0 solves the reduced problem Q(U0 ) = 0 or U0 = E(u), ' & d P I (U0 ) U0t + Aj (U0 )U0xj = 0,
(22)
j=1
˜ +∞). U0 (x, 0) = I(x, Here P I (U0 ) = P I (resp. P II ) denotes the (n − r) × n (resp. r × n) matrix consisting of the first (n − r) (resp. last r) rows of P (U0 ). The system of equations in (22) is the equilibrium system. Under the second stability condition, we use energy methods to prove Theorem 4.3 Suppose the system (2) satisfies the second stability condition, ¯ =U ¯ (x) satisfies U ¯ (x) ∈ G0 ⊂⊂ G for all x and U ¯ ∈ Hs the initial value U with an integer s > d/2 + 1, and the assumptions (a) and (b) concerning the limiting inner problem and the equilibrium manifold hold. Then there are two
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positive constants T∗ and K, independent , of , such- that all terms Uk , Ik (k = 0, 1, · · · , m) in (20) can be found in C [0, T∗ ], H s+1 , Um (x, t) takes values in a convex compact subset of G, the H s -solution U exists in the uniform time interval [0, T∗ ], and ||U (t) − Um (t)||s ≤ Km+1/2 for sufficiently small and t ∈ [0, T∗ ]. This theorem is only for the simple case, where the coefficients in (2) are independent of (x, t, ), the initial data are periodic and independent of , and the second stability condition holds. The interested reader may consult [Yo92] for the more general cases, where the second stability condition is replaced with the first one. The latter consists of (i), (ii) (of the second stability condition) and (iii)’. the hyperbolic part and the source term are coupled in the following fashion A0 (U )QU (U ) + Q∗U (U )A0 (U ) ≤ 0
for
U ∈ E.
A direct consequence of Theorem 4.3 is U (x, t) =
m
k Uk (x, t) +
k=0
m
k Ik (x, t/) + O(m+1/2 )
(23)
k=0
for x ∈ Rd and t ∈ [0, T∗ ]. In particular, it follows from (23) and (21) that, out of the initial-layer, the solution U of the original problem (2) converges to the unique smooth solution U0 of the reduced problem (22) as goes to zero. The proof of Theorem 4.3 can be found in [Yo92, Yo99a]. Let us remark that our approach is valid only for smooth solutions but works also for other singular limit problems (see [LaY01, JY03, Yo04]). One advantage of this approach is that it allows us to characterize the exact limiting behaviors and to obtain the convergence rate, as shown by the expansion (23).
5 Shock structure problems This section is devoted to the existence of traveling wave solutions to the relaxation system (3) satisfying the second stability condition. We assume that q(u, v) = 0 uniquely determines v in term of u, say v = h(u). Thus, the equilibrium manifold can be expressed with u E(u) = h(u)
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and the equilibrium system in (22) becomes a closed system of conservation laws d ut + fj (u, h(u))xj = 0. (24) j=1
Note that P (U ) in the stability condition can be simply taken to be 0 In−r P (U ) = . qv−1 qu (U ) Ir For conservation laws (24), the simplest discontinuous solutions are of the form 0 u− , if ξ · x < s∗ t; u(x, t) = (25) otherwise. u+ , Here the quantities u± , s∗ and ξ = (ξ1 , ξ2 , · · · , ξn ) are given and satisfy the Rankine-Hugoniot relation s∗ (u+ − u− ) =
d
ξj [fj (u+ , h(u+ )) − fj (u− , h(u− ))].
j=1
Suppose the characteristic matrix j ξj fj (u, h(u))u of (24) has an isolated eigenvalue λk (u). The piecewise constant weak solution (25) is called a k-th shock-front if it satisfies Liu’s strict entropy condition [Li76]: s∗ = sk (ρ+ ) < sk (ρ)
(26)
for all ρ strictly between 0 and ρ+ . Here ρ parametrizes the k-th Hugoniot curve u(ρ) = u(ρ, u− ) passing through u− = u(0), u+ = u(ρ+ ), and sk (ρ) is uniquely determined by sk (0) = λk (u− ) and the Rankine-Hugoniot relation sk (ρ)(u(ρ) − u− ) =
d
ξj [fj (u(ρ), h(u(ρ))) − fj (u− , h(u− ))].
j=1
The question here is when the relaxation system (3) has a smooth traveling wave solution (u , v )(x, t) = Φ((ξ ·x−s∗ t)/) converging to (u, h(u)) as goes to zero, where u is the discontinuous solution (25). Set ω = (ξ · x − s∗ t)/. Such a Φ(ω) should satisfy −s∗ Φω +
2 d j=1
3 ξj Fj (Φ) = Q(Φ), ω
(27)
Φ(±∞) = E(u± ) for ω ∈ (−∞, ∞). Such a traveling wave solution is called a relaxation shock profile and its existence is of independent interest physically (see
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[BR98, MR98]). We recall that the analogous question was asked by Gelfand for viscosity approximations of hyperbolic conservation laws and was satisfactorily answered by Majda and Pego in [MP85]. For the case where r = 1, (27) can be directly reduced to a one-dimensional connection problem, for which a general existence theorem is well known. With n = 2, this case was considered by Liu in [Li87] and it was shown there that the existence follows from the well-known subcharacteristic condition imposing on (3). As shown in [Yo01], this subcharacteristic condition has various generalizations for (3) with general n and r. Our interest is in the general case: r ≥ 1. The result is Theorem 5.1 Assume (3) satisfies the second stability condition, and λk (u− ) with u− fixed is an isolated eigenvalue for the equilibrium system (24) but not an eigenvalue for the relaxation system (3). Then the following three statements are true: (a) if u+ = u− and s∗ = sk (ρ) for some ρ strictly between 0 and ρ+ , then there is no smooth profile Φ(ω) of (27) close to E(u− ); (b) if s∗ < sk (ρ) for all ρ strictly between 0 and ρ+ , then there exist δ1 > 0 and δ2 > 0 so that, for any u+ satisfying |u+ − u− | < δ1 and lying in the k-th Hugoniot curve for u− with some speed s∗ , a unique smooth trajectory Φ(ω) of (27) satisfying |Φ(ω) − E(u− )| < δ2 exists connecting E(u− ) from left to E(u+ ) at the right; (c) if s∗ > sk (ρ), then there exist δ1 > 0 and δ2 > 0 so that, for any u+ satisfying |u+ − u− | < δ1 and lying in the k-th Hugoniot curve for u− with some speed s∗ , a unique smooth trajectory Φ(ω) of (27) satisfying |Φ(ω) − E(u− )| < δ2 exists connecting E(u− ) from right to E(u+ ) at the left. Moreover, if s∗ = λk (u± ) then Φ(ω) approaches E(u± ) exponentially as ω → ±∞. The proof of Theorem 5.1 was motivated by the work of Majda and Pego [MP85]. Indeed, when r > 1, (27) is a genuinely multi-dimensional connection problem, for which we do not know a general existence theorem. As in [MP85] for viscosity problems, our strategy is to use the standard technique in dynamic bifurcation theory to reduce (27) to a one-dimensional problem. The possibility of this reduction is guaranteed by the preconditions in Theorem 5.1. Note that (3) is structurally different from the viscosity approximations and thus needs a delicate analysis of its algebraic structure. Moreover, our proof involves a new parametrization of Hugoniot curves. The details can be found in [YZ00].
6 Entropy and global solutions This section is based on [Yo04a]. Notice that, without the source term Q(U ), (3) reduces to a system of conservation laws. In that case, it is well known
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that solutions generally develop singularities (e.g. shock waves) in finite time, even when the initial data are smooth and small [Da00]. In [Yo04a], we presented a set of structural conditions under which source terms may prevent the breakdown of smooth solutions. Namely, we developed a existence theory of global smooth solutions for balance laws (3). Let Ue ∈ E. The structural conditions read as follows. (i.) The Jacobian qv (Ue ) is invertible. (ii.) There is a strictly convex smooth function η(U ), defined in a convex compact neighborhood G of Ue , such that ηU U (U )FjU (U ) is symmetric for all U ∈ G and all j. (iii.) There is a positive constant cG such that for all U ∈ G, [ηU (U ) − ηU (Ue )]Q(U ) ≤ −cG |Q(U )|2 . , (iv.) The kernel of the Jacobian QU (Ue ), Ker QU (Ue ) , contains no eigenvector of the matrix j ωj FjU (Ue ) for any ω = (ω1 , ω2 , · · · , ωd ) ∈ Sd−1 (the unit sphere in Rd ). Here and below, the subscripts v, U, · · · denote the corresponding partial derivatives with respect to these variables, so ηU U (U ) is the Hessian matrix of η(U ). About these conditions, we make several comments. Together, (i)–(iii) constitute an entropy dissipation condition for balance laws (3). Alternative conditions in the same spirit were previously introduced by Chen, Levermore and Liu [CLL94], M¨ uller, Ruggeri and Boillat [MR98, BR98], and Yong [Yo01], with the aim of studying the zero relaxation limit. However, these conditions do not seem to provide the proper setting for proving global existence. See [Ze99] for a similar comment on the Chen-Levermore-Liu condition. Further comments need some consequences of Condition (i). Introduce u ≡ V (U ) (28) V = q(U ) and set Ve = V (Ue ). Since qv (Ue ) is invertible, V = V (U ) has an inverse U = U (V ) for U close to Ue . In particular, v can be viewed as a function v(u, q) of u and q. Set Aj (V ) = UV−1 (V )FjU (U (V ))UV (V ). Then, for smooth solutions, (3) can be rewritten as Vt +
d
, Aj (V )Vxj = diag 0, qv (U ) V,
(29)
j=1
which is called the normal form of the balance law (3). Note that the transformation (28) does not destroy the conservative form of the u-equation in (3):
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ut +
d
fj (u, v(u, q))xj = 0.
j=1
The normal form (29) of (3) enables us easily to use the Chapman-Enskog expansion (or the Maxwell iteration [MR98]) to derive a first-order approximation to our system. With the partition in Section 2, we rewrite the q-equation in (29) as 2 3 22 A21 (V )u + A (V )q q = qv−1 (U (V )) qt + , x x j j j j j
j
with a positive parameter 0 =1, for the present problem $ 1, for the zero relaxation limit problem, and then iterate once to obtain q = qv−1 (U (u, 0))
2 A21 j (u, 0)uxj + O( ).
j
Substituting the truncation into the u-equation in (29), we arrive at ! fjv (U (u, 0))qv−2 (U (u, 0))A21 fj (U (u, 0))xj + ut + k (u, 0)uxk x = 0. j
j
jk
(30) Thus, we have a short and logically simple version of the classical ChapmanEnskog theory for balance laws [CLL94]. Note that no entropy function is involved here. In [Yo04a] we proved Theorem 6.1 Assume the entropy dissipation condition (i)–(iii) holds. Then UV∗ ηU U UV evaluated at U = Ue is a block-diagonal matrix (under the aforementioned partition) and there is a positive definite r × r-matrix Λ(Ue ) such that 0 0 VU (Ue ). ηU U (Ue )QU (Ue ) + Q∗U (Ue )ηU U (Ue ) = −VU∗ (Ue ) 0 Λ(Ue ) By this theorem, the present entropy dissipation condition implies the second stability condition. Thus, it provides a proper setting for studying the zero relaxation limit as well as the existence and stability of relaxation shock profiles [Zu01, MZ02]. Moreover, as was shown in [Yo04a, Yo01], our entropy dissipation condition is satisfied by many equations of classical physics of the form (3). Condition (iv), which is generally referred to as the “Kawashima condition”, was first formulated by Shizuta and Kawashima [SK85] for symmetric
On Hyperbolic Relaxation Problems
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hyperbolic-parabolic systems of the form (30) with source term, in order to derive decay estimates on solutions. Unlike these authors, we do not require that ηU U (Ue )QU (Ue ) is symmetric. In [Ka84, SK85, UKS85], Kawashima et al. have verified that this condition is satisfied by many classical systems of hyperbolic-parabolic type governing fluid flow, in particular by the Navier-Stokes equations. At the same time, we have seen that such hyperbolic-parabolic systems can be derived as firstorder approximations to hyperbolic systems of balance laws, of the form (3), through the Chapman-Enskog expansion. Responding to this connection, we proved Theorem 6.2 Under the entropy dissipation condition (i)–(iii), a system of balance laws (3) satisfies the Kawashima condition if and only if its first-order approximation (30) satisfies it. Now, we turn to state the global existence result. Theorem 6.3 Let s ≥ s0 +1 ≡ [d/2]+2 be an integer and Conditions (i)–(iv) hold at Ue ∈ E. Then there are two constants c1 , c2 such that if U0 = U0 (x) ∈ Ue + H s (Rd ) satisfies U0 − Ue s ≤ c1 , then the system of balance laws (3) with U0 as its -initial value has a unique , global solution U = U (x, t) ∈ Ue + C 0, ∞; H s (Rd ) satisfying U (·, T ) −
Ue 2s
T
Q(U )(·, t)2s dt
+ 0
T
∇U (·, t)2s−1 dt ≤ c2 U0 − Ue 2s
+ 0
(31) for any T > 0. 2 d In this theorem, H s,(Rd ) denotes the - usual L -Sobolev space of order s on R , s d its norm is · s , C 0, ∞; H (R ) denotes the space of continuous functions on [0, ∞) with values in H s (Rd ), A(·, t)s is the norm of A = A(x, t) taken with respect to x while t is viewed as a parameter, and ∇ is the gradient operator with respect to x. Note that the estimate (31) implies # , -# as t → ∞, sup #∂ α U (x, t) − Ue # → 0, x∈Rd
for all multi-indices α satisfying |α| ≤ s − s0 − 1. Here we should mention that a similar existence result has been obtained independently by Hanouzet and Natalini [HN03] for one-dimensional problems. Moreover, conditions of the form (i)–(iv) have been used by Ruggeri and Serre in [RS04] to show that constant equilibrium states are timeasymptotically L2 -stable in a certain class of weak entropy solutions, again for one-dimensional problems. Theorem 6.3 is proved by the standard continuation argument together with the local-in-time existence theory for symmetrizable hyperbolic systems
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(see, e.g. [Ma84]) and is based on the a priori estimate (31). This estimate is derived by the energy method. For that purpose, we employ the normal form (29) of (3), together with Theorem 6.1 expressing a basic consequence of Conditions (i)–(iii). Moreover, we apply the following skew-symmetrizer theorem, in Fourier space, due to Shizuta and Kawashima [SK85]: Theorem 6.4 (Shizuta-Kawashima, 1985) Assume (i) and (ii) hold. Condition (iv) is equivalent to the existence of a constant cS > 0 and a skewsymmetric real matrix K = K(ω) ∈ C ∞ (Sd−1 ) satisfying K(−ω) = −K(ω) and K(ω)A(ω) − A∗ (ω)K(ω) ≥ 2cS In − 2 diag(0, Ir ), for any ω ∈ Sd−1 . The above global existence theory generalizes that due to Kawashima [Ka83] for discrete velocity Boltzmann equations and can be used to many other equations of classical physics. As a new application, we consider discrete velocity BGK models constructed in [AN00, Bo99] to approximate hyperbolic systems of conservation laws: ut +
d
gj (u)xj = 0,
(32)
j=1
where u ∈ U ⊂ Rp (a convex open set) and gj maps U to Rp . The BGK models are of the form fkt + a(k) · ∇x fk = Mk (u) − fk for k ∈ Ξ := {1, 2, · · · , N }. Here fk = fk (x, t) ∈ Rp is unknown, a(k) ∈ Rd is a constant vector with components aj (k) (j = 1, 2, · · · , d), a(k) · ∇x = d N p j=1 aj (k)∂xj , u = k=1 fk , and Mk : U → R satisfies the consistency relations N k=1
Mk (u) = u,
N
aj (k)Mk (u) = gj (u),
∀u ∈ U.
(33)
k=1
Clearly, a BGK model is determined by choosing the a(k)’s and Mk (u)’s so that the consistency relations (33) hold. In [Bo99], the Mk (u)’s are taken as linear combinations of u and the gj (u)’s: Mk (u) = αk0 u +
d
αkj gj (u).
j=1
The coefficients αkj (j = 0, 1, · · · , d) are chosen so that the consistency re d lations (33) hold and the matrix αk0 Ip + j=1 αkj gju (u) has only positive eigenvalues.
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With the Mk (u)’s thus obtained, Conditions (i)–(iv) were verified in [Yo04a] with the entropy function η = η(U ) constructed in [Se00a, Bo99], provided that (a) the approximated conservation laws (32) are endowed with a strictly convex entropy function η˜ = η˜(u); (b) the a(k)’s are chosen so that the d × (N − 1)-matrix ! a(2) − a(1), a(3) − a(1), · · · , a(N ) − a(1) has rank d. Finally, we remark that, unlike the entropy dissipation condition (i)–(iii), (iv) does not hold for all physical models. An example is the system of equations of gas dynamics in thermal non-equilibrium (see [Ze99, Yo01]). Even so, a global existence theorem was established in Zeng [Ze99] for this model. This indicates that Condition (iv) is not quite necessary for global existence. It would be interesting to weaken that condition while preserving the global existence.
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[Ce88] [CLL94]
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Appendix: Color Plates
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Appendix: Color Plates
Plate 1: (Fig. 2 on page 38) HAT firing in the laboratory, 2000 A, 0.8 g/s.
Plate 2: (Fig. 4 on page 39) HAT: Heavy particle temperature Th, 2000 A, 0.8 g/s.
Appendix: Color Plates
Plate 3: (Fig. 7 on page 41) HAT: Density logIo(p), 4000 A, 0.8 g/s.
Plate 4: (Fig. 8 on page 41) HAT: Density log,,(p), 5000 A, 0.8 g/s.
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Appendix: Color Pla.tes
Plate 5: (Fig. l1 on page 43) DT2: Density log,,(p), 4000 A, 0.8 g/s.
Plate 6: (Fig. 12 on page 43) DT2: Density log,,(p), 5000 A, 0.8 g/s.
Appendix: Color Plates
525
Plate 7: (See also Fig. 9 on page 66) Density plot of a supersonic flow propagating into a channel.
Plate 8: (See also Fig. 10 on page 67) Density plot of a thermally driven flow.
526
Appendix: Color Plates
Plate 9: (Fig. 10 on page 89) Color plots of the temperature and schlieren plots of the density on refinement regions in the first (left) and second half (right) of a detonation cell.
Plate 10: (Fig. 11 on page 89) Schlieren plots of ρ (upper row) and YOH (lower row) in the first (left) and second (right) half of detonation cell, mirrored at x2 = 0 cm, 5.0 cm < x1 < 7.0 cm. The plots of YOH are overlaid by a blue isosurface of ρ that visualizes lig .
Appendix: Color Plates
Plate 11. to be continued
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Appendix: Color Plates sequel to Plate 11.
Plate 11: (Fig. 7 on page 126) Time evolution of magnetic field (field lines are shown in black) and temperature (color coded) from a 2D simulation run of solar magneto-convection near the visible solar surface (located at z 100 km height). A homogeneous vertical field of 100 Gauss has been introduced at t = 0 after a statistically stationary convection pattern has evolved. Within a few minutes, most of the magnetic flux is transported by the converging horizontal flows to the cool downflow region (flux expulsion). Owing to the suppression of the convective energy transport, the gas in the flux concentrations cools and sinks; lateral compression by the external gas pressure then leads to a strong intensification of the field strength, which reaches kilogauss values. About 20 minutes after the introduction of the magnetic field, the flux concentrations have merged into three large flux sheets (labeled I,II,III), which start to determine the surrounding flow pattern with strong downflows surrounding the flux sheets. After about 30 minutes a quasi-stationary situation has developed. The velocity field in this state is shown in the form of velocity vectors in the last panel.
Appendix: Color Plates
Plate 12: (Fig. 12 on page 158) Expansion of a liquefaction shock.
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Appendix: Color Plates
Plate 13. to be continued
Appendix: Color Plates sequel to Plate 13.
Plate 13. to be continued
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Appendix: Color Plates sequel to Plate 13.
Plate 13: (Fig. 13 on page 161) Bubble collapse.
Appendix: Color Plates
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Plate 14: (Fig. 9 on page 198) Results using the mixed GLM-MHD scheme for a 2d magnetic flux tube simulation, on the left with the Bgfix modification, on the right using only the base scheme. The B, component is shown in a small region of the domain (top) and a color representation of the grid refinement (bottom), red indicating a very high grid resolution and blue a very coarse grid. With the Bgfix correction the grid consists of about 12000, without the correction of about 32000 elements. Note also the loss of symmetry caused by the base scheme.
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Appendix: Color Plates
(a) Entropy isosurface and levelsurface in the central part of the domain (left) and the grid partitioning and refinement (right) at times t = 0.0, 6.0, 9.0, and t = 12.0.
simulation time 70
execution time
60 50 40 30 20 10 0 0
1
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3
no. of elements, maximum: 6687440 minimum runtime for numerics average runtime for numerics
4
5
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8
maximum runtime for numerics runtime, total: 233928 s
(b) Effect of the load balancing on the runtime. Top: grid partitioning, refinement, and relative load of partitions at t = 6.2, 6.6, 6.65. Bottom: graphs of runtime. Load balancing takes place about every 200 steps (peaks in runtime graph) and requires approximately the runtime of three normal time steps. For example, load balancing occurs shortly before t = 6.2. Due to grid refinement we observe an increase of the load, e.g., for the light green and the brown partition between t = 6.2 and t = 6.6. Thus the maximal and minimal runtime diverge. Consequently, the total runtime increases more than it would be justified by the growth of the number of elements. Following our load balancing strategy the grid is therefore repartitioned between t = 6.6 and t = 6.65. On average the total runtime increases by the same amount as the number of elements in the grid. Note that this is the optimal behaviour for an explicit finite volume scheme. Plate 15: (Fig. 11 on page 199) Simulation of an exploding flux tube in 3d on 8 processors of an IBM-SP2.
Appendix: Color Plates
Plate 16: (Fig. 7 on page 251) Initialization of a 3D bow shock with shock, subset grid, inner sphere and constant pressure values.
Plate 18: (Fig. 9 on page 251) Snapshot of an inviscid bow shock with Mach=3 and three cylinders. Pressure values are plotted
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Plate 17: (Fig. 8 on page 251) Final state of the 3D bow shock calculation with front, subset mesh and pressure values in the cut plane.
Plate 19: (Fig. 10 on page 251) Zoom of area between both lower cylinders; pressure values, front, subset mesh and velocity vectors.
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Appendix: Color Plates
Plate 20: (Fig. 11 on page 252) Snapshot of an inviscid unstable 2D CJ-detonation with pressure values calculated on an unstructured grid.
Plate 21: (Fig. 12 on page 252) Masterslave combination for a restricted recompression shock wave on a transonic airfoil: Colors of constant Mach number and front position.
Appendix: Color Plates
337
Plate 22: (Fig. 3 on page 329) Spatial appearance of the number of dust particles (lognd [cmP3]:false color background) and the vorticity (V x w: black and grey contour lines) of the 2D velocity field for t = 0.8s of a simulation with Tref= 2100 K. pref = 3.16 l O - ' g ~ m - ~ .vref = CS.
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Appendix: Color Plates
Plate 23: (Fig. 12 on page 381) Flame propagation into quiescent fuel at ρu = 5 × 107 g cm−3 , resolution: 200 × 200 cells, snapshots taken at (a) 0, (b) 7.5, (c) 15, and (d) 30 growth times τLD of a perturbation with λ = 3.2 × 104 cm. Color-coded is the vorticity of the flow field.
Appendix: Color Plates
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Plate 24: (Fig. 13 on page 382) Flame propagation into vortical fuel at ρu = 5×107 g cm−3 ; velocity fluctuations at the right boundary: v /ulam = 2.5; resolution: 300 × 200 cells; snapshots taken at (a) 0 s, (b) 8.0 × 10−3 s, (c) 1.6 × 10−2 s, and (d) 4.8 × 10−2 s. Color-coded is the vorticity of the flow field.
0.9
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Ex 43.7 38.2 32.6 27.0 21.4 15.9 10.3 4.7 -0.9 -6.5 -12.0 -17.6 -23.2 -28.8 -34.4
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y
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Ex 21.2 19.5 17.9 16.2 14.5 12.8 11.1 9.4 7.7 6.0 4.3 2.6 0.9 -0.7 -2.4
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Plate 25: (Fig. 1 on page 395) Snapshots of the E1 field component obtained
with uncorrected (left) and corrected (right) computation.
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Appendix: Color Plates
Plate 26. to be continued
Appendix: Color Plates sequel to Plate 26.
Plate 26. to be continued
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Appendix: Color Plates sequel to Plate 26.
Plate 26: (Fig. 11 on page 426) Three-dimensional propagation of a premixed flame over an obstacle: Representation of the isoline G=0 as well as the velocity field (vector arrows); local burning speeds are modeled over G-LEM flame modules.
Plate 27: (Fig. 12 on page 426) Two-dimensional cut plane (x-y-direction) through the flow field: The velocity field (vector arrows) and the turbulent kinetic energy are shown.
