17
Eleuterio F. Toro Riemann Solvers and Numerical Methods for Fluid Dynamics
Eleuterio F. Toro
Riemann Solvers and Numerical Methods for Fluid Dynamics
Springer-Verlag Berlin Heidelberg GmbH
Eleuterio F. Toro
Riemann Solvers and Numerical Methods for Fluid Dynamics
A Practical Introduction
With 223 Figures
~Springer
Professor Dr. ELEUTERIO F. ToRo
Manchester Metropolitan University Dept. of Computing and Mathematics Chester Street Manchester MI sGD United Kingdom
E P Boden. CHIMERA/AMR meshes and WAF method with HLLC Riemann solver.
Library of Congress Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek- CIP-Einheitsaufnahme
Toro, Eleuterio F.: Riemann solvers and numerica! methods for fluid dynamics: a practica! introduction 1 Eleuterio F. Toro.
ISBN 978-3-662-03492-7 ISBN 978-3-662-03490-3 (eBook) DOI 10.1007/978-3-662-03490-3
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, broadcasting, reproduction on microfilm or in other way, and storage in data banks. Duplication of this publicat ion 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-Verlag Berlin Heidelberg GmbH.
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© Springer-Verlag Berlin Heidelberg 1997 Originally published by Springer-Verlag Berlin Heidelberg New York in 1997 Softcover reprint of the hardcover 1st edition 1997
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.
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Preface
In 1917, the British scientist L. F. Richardson made the first reported attempt to predict the weather by solving partial differential equations numerically, by hand! It is generally accepted that Richardson's work, though unsuccessful, marked the beginning of Computational Fluid Dynamics (CFD), a large branch of Scientific Computing today. His work had the four distinguishing characteristics of CFD: a PRACTICAL PROBLEM to solve, a MATHEMATICAL MODEL to represent the problem in the form of a set of partial differential equations, a NUMERICAL METHOD and a COMPUTER, human beings in Richardson's case. Eighty years on and these four elements remain the pillars of modern CFD. It is therefore not surprising that the generally accepted definition of CFD as the science of computing numerical solutions to Partial Differential or Integral Equations that are models for fluid flow phenomena, closely embodies Richardson's work.
COMPUTERS have, since Richardson's era, developed to unprecedented levels and at an ever decreasing cost. PRACTICAL PROBLEMS to solved numerically have increased dramatically. In addition to the traditional demands from Meteorology, Oceanography, some branches of Physics and from a range of Engineering Disciplines, there are at present fresh demands from a dynamic and fast-moving manufacturing industry, whose traditional build-test-fix approach is rapidly being replaced by the use of quantitative methods, at all levels. The need for new materials and for decision-making under environmental constraints are increasing sources of demands for mathematical modelling, numerical algorithms and high-performance computing. MATHEMATICAL MODELS have improved, though the basic equations of continuum mechanics, already available more than a century before Richardson's first attempts at CFD, are still the bases for modelling fluid flow processes. Progress is required at the level of Thermodynamics, equations of state, and advances into the modelling of non-equilibrium and multiphase flow phenomena. NuMERICAL METHODS are perhaps the success story of the last eighty years, the last twenty being perhaps the most productive. This success is firmly based on the pioneering works of scientists such as von Neumann, whose research on stability explained and resolved the difficulties experienced by Richardson. This success would have been impossible without the contributions from Courant, Friedrichs, Richtmyer, Lax, Oleinik, Wendroff, Godunov, Rusanov,
VI
van Leer, Harten, Roe, Osher, Colella, Yee, and many others. The net result is: more accurate, more efficient, more robust and more sophisticated numerical methods are available for ambitious practical applications today.
Due to the massive demands on CFD and the level of sophistication of numerical methods, new demands on education and training of the scientists and engineers of the present and the future have arisen. This book is an attempt to contribute to the training and education in numerical methods for fluid dynamics and related disciplines.
The contents of this book were developed over a period of many years of involvement in research on numerical methods, application of the methods to solve practical problems and teaching scientist and engineers at the post-graduate level. The starting point was a module for a Masters Course in Computational Fluid Dynamics at the College of Aeronautics, Cranfield, UK. The material was also part of short courses and lectures given at Cranfield UK; the Ernst Mach Institute, Freiburg, Germany; the Shock Wave Research Centre, Tohoku University, Sendai, Japan; the Department of Mathematics and the Department of Civil and Environmental Engineering, University of Trento, Italy; the Department of Mathematics, Technical University Federico Santa Maria, Chile; the Department of Mechanics, Technical University of Aachen, Germany; and the Manchester Metropolitan University (MMU), Manchester, UK.
This book is about modern shock-capturing numerical methods for solving time-dependent hyperbolic conservation laws, with smooth and discontinuous solutions, in general multidimensional geometries. The approach is comprehensive, practical and, in the main, informal. All necessary items of information for the practical implementation of all methods studied here, are provided in detail. All methods studied are illustrated through practical numerical examples; numerical results are compared with exact solutions and in some cases with reliable experimental data.
Most of the book is devoted to a coherent presentation of Godunov methods. The developments of Godunov's approach over the last twenty years have led to a mature numerical technology, that can be utilised with confidence to solve practical problems in established, as well as new, areas of application. Godunov methods rely on the solution of the Riemann problem. The exact solution is presented in detail, so as to aid the reader in applying the solution methodology to other hyperbolic systems. We also present a variety of approximate Riemann solvers; again, the amount of detail supplied will hopefully aid the reader who might want to apply the methodologies to solve other problems. Other related methods such as the Random Choice Method and the Flux Vector Splitting Method are also included. In addition, we study centred (non-upwind) shock-capturing methods. These schemes are much less sophisticated than Godunov methods, and offer a cheap and simple alternative. High-order extensions of these methods are constructed for scalar PDEs, along with their Total Variation Diminishing (TVD) versions. Most
VII
of these TVD methods are then extended to one-dimensional non-linear systems. Techniques to deal with PDEs with source terms are also studied, as are techniques for multidimensional systems in general geometries.
The presentation of the schemes for non-linear systems is carried out through the time-dependent Euler equations of Gas Dynamics. Having read the relevant chapters/sections, the reader will be sufficiently well equipped to extend the techniques to other hyperbolic systems, and to convectionreaction-diffusion PDEs.
There are at least two ways of utilising this book. First, it can be used as a means for self-study. In the presentation of the concepts, the emphasis has been placed on clarity, sometimes sacrifying mathematical rigour. The typical reader in mind is a graduate student in a Department of Engineering, Physics, Applied Mathematics or Computer Science, embarking on a research topic that involves the implementation of numerical methods, from first principles, to solve convection-reaction-diffusion problems. The contents of this book may also be useful to numerical analysts beginning their research on algorithms, as elementary background reading. Such users may benefit from a comprehensive self-study of all the contents of the book, in a period of about two months, perhaps including the practical implementation and testing of most numerical methods presented. Another class of readers who may benefit from self-studying this book are scientists and engineers in industry and research laboratories. At the cost of some repetitiveness, each chapter is almost self-contained and has plenty of cross-referencing, so that the reader may decide to start reading this book in the middle or jump to the last chapter.
This book can also be used as a teaching aid. Academics involved in the teaching of numerical methods may find this work a useful reference book. Selected chapters or sections may well form the bases for a final year undergraduate course on numerical methods for PDEs. In a Mathematics or Computer Science Department, the contents may include: some sections of Chap. 1, Chaps. 2, 5, 13, some sections of Chap. 14, Chap. 15 and some sections of Chap. 16. In a Department of Engineering or Physics, one may include Chaps. 3, 4, 6, 7, 8, 9, 10, 11, 12, 15, 16 and 17. A postgraduate course may involve most of the contents of this book, assuming perhaps a working knowledge of compressible fluid dynamics. Short courses for training engineers and scientists in industry and research laboratories can also be based on most of the contents of this book.
Tito Toro Manchester, UK March 1997.
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ACKNOWLEDGEMENTS
The author is indebted to many colleagues who over the years have kindly arranged short and extended visits to their establishments, have organised seminars, workshops and short courses. These events have strongly influenced the development of this book. Special thanks are due to Dr. W. Heilig (Freiburg, Germany); Professors V. Casulli and A. Armanini (Trento, Italy); Professor K. Takayama (Sendai, Japan); Professor J. Figueroa (Valparaiso, Chile) and Professor J. BaUmann (Aachen, Germany). The final stages of the typing of the material were carried out while the author was at MMU, Manchester, UK; the support provided is gratefully acknowledged. Thanks are also due to my former and current PhD students, for their comments on my lectures, on the contents of this book and for their contribution toresearch in relevant areas. Special thanks are due to Stephen Billett (Centre for Atmospheric Science, University of Cambridge, UK), Caroline Lowe (Department of Aerospace Science, Cranfield University, UK), Nikolaos Nikiforakis (DAMTP, University of Cambridge, UK), Ed Boden (Cranfield and MMU), Ms Wei Hu, Mathew Ivings and Richard Millington (MMU). Thanks are due to my son Marcelo, who helped with the type-setting, and to David Ingram and John Nuttall for their help with the intricacies of Latex. The author thanks Sringer-Verlag, and specially Miss Erdmuthe Raufelder in Heidelberg, for their professional assistance.
The author gratefully acknowledges useful feedback on the manuscript from Stephen Billett (University of Cambridge, UK), John Clarke (Cranfield University, UK), Jean-Marc Moschetta (SUPERO, Toulouse, France), ClausDieter Munz (University of Stuttgart, Germany), Jack Pike (Bedford, UK), Ning Qin (Cranfield University, UK), Tsutomu Saito (Cray Japan), Charles Saurel (University of Provence, France), Trevor Randall (MMU, Manchester, UK), Peter Sweby (University of Reading, UK), Marcelo Toro (IBM Glasgow, UK), Helen Yee (NASA Ames, USA) and Clive Woodley (DERA, Fort Halstead, UK).
The permanent encouragement from Brigitte has been an immensely valuable support during the writing of this book. Thank you Brigitte!
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To my children
MARCELO
CARLA
VIOLETA
and
EVITA
Table of Contents
Preface 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 V
1. The Equations of Fluid Dynamics 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
lol The Euler Equations 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2
1.1.1 Conservation-Law Form 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3
1.1.2 Other Compact Forms 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4
1.2 Thermodynamic Considerations 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5
1.201 Units of Measure 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5
1.202 Equations of State (EOS) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1.203 Other Variables and Relations 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7
1.204 Ideal Gases 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 1.205 Covolume and van der Waal Gases 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13
1.3 Viscous Stresses 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15
1.4 Heat Conduction 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17
1.5 Integral Form of the Equations 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18
1.501 Time Derivatives 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19
1.502 Conservation of Mass 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20
1.503 Conservation of Momentum 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21
1.5.4 Conservation of Energy 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23
1.6 Submodels 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25
1.601 Summary of the Equations 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25
1.602 Compressible Submodels 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 27
1.603 Incompressible Submodels 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 33
2. Notions on Hyperbolic Partial Differential Equations 0 0 0 0 0 41
201 Quasi-Linear Equations: Basic Concepts 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 41
202 The Linear Advection Equation 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 47
20201 Characteristics and the General Solution 0 0 0 0 0 0 0 0 0 0 0 0 47
20202 The Riemann Problem 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 49
203 Linear Hyperbolic Systems 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 50
20301 Diagonalisation and Characteristic Variables 0 0 0 0 0 0 0 0 0 51
20302 The General Initial-Value Problem 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 52
20303 The Riemann Problem 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 55
203.4 The Riemann Problem for Linearised Gas Dynamics 0 0 58
XII Table of Contents
2.3.5 Some Useful Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.4 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.4.1 Integral Forms of Conservation Laws................ 61 2.4.2 Non-Linearities and Shock Formation . . . . . . . . . . . . . . . 65 2.4.3 Characteristic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.4.4 Elementary-Wave Solutions of the Riemann Problem . 83
3. Some Properties of the Euler Equations . . . . . . . . . . . . . . . . . . 87 3.1 The One-Dimensional Euler Equations.................... 87
3.1.1 Conservative Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.1.2 Non-Conservative Formulations . . . . . . . . . . . . . . . . . . . . 91 3.1.3 Elementary Wave Solutions of the Riemann Problem.. 94
3.2 Multi-Dimensional Euler Equations ....................... 102 3.2.1 Two-Dimensional Equations in Conservative Form ... 103 3.2.2 Three-Dimensional Equations in Conservative Form .. 107 3.2.3 Three-Dimensional Primitive Variable Formulation ... 108 3.2.4 The Split Three-Dimensional Riemann Problem ...... 110
3.3 Conservative Versus Non-Conservative Formulations ........ 111
4. The Riemann Problem for the Euler Equations ........... 115 4.1 Solution Strategy ....................................... 116 4.2 Equations for Pressure and Particle Velocity ............... 119
4.2.1 Function A for a Left Shock ....................... 120 4.2.2 Function h for Left Rarefaction ................... 122 4.2.3 Function fR for a Right Shock ..................... 123 4.2.4 Function fR for a Right Rarefaction ................ 124
4.3 Numerical Solution for Pressure .......................... 125 4.3.1 Behaviour of the Pressure Function ................. 125 4.3.2 Iterative Scheme for Finding the Pressure ........... 127 4.3.3 Numerical Tests .................................. 129
4.4 The Complete Solution .................................. 133 4.5 Sampling the Solution ................................... 136
4.5.1 Left Side of Contact: S = xjt::::; u* ................. 137 4.5.2 Right Side of Contact: S = x/t 2: u* ................ 137
4.6 The Riemann Problem in the Presence of Vacuum .......... 138 4.6.1 Case 1: Vacuum Right State ....................... 140 4.6.2 Case 2: Vacuum Left State ........................ 141 4.6.3 Case 3: Generation of Vacuum ..................... 142
4. 7 The Riemann Problem for Covolume Gases ................ 143 4. 7.1 Solution for Pressure and Particle Velocity ........... 143 4.7.2 Numerical Solution for Pressure .................... 147 4.7.3 The Complete Solution ............................ 147 4. 7.4 Solution Inside Rarefactions ....................... 148
4.8 The Split Multi-Dimensional Case ........................ 149 4.9 FORTRAN 77 Program for Exact Riemann Solver .......... 151
Table of Contents XIII
5. Notions on Numerical Methods ........................... 159 5.1 Discretisation: Introductory Concepts ..................... 159
5.1.1 Approximation to Derivatives ...................... 160 5.1.2 Finite Difference Approximation to a PDE .......... 161
5.2 Selected Difference Schemes .............................. 163 5.2.1 The First Order Upwind Scheme ................... 164 5.2.2 Other Well-Known Schemes ....................... 168
5.3 Conservative Methods .................................. 170 5.3.1 Basic Definitions ................................. 171 5.3.2 Godunov's First-Order Upwind Method ............. 173 5.3.3 Godunov's Method for Burgers's Equation ........... 176 5.3.4 Conservative Form of Difference Schemes ............ 179
5.4 Upwind Schemes for Linear Systems ...................... 183 5.4.1 The CIR Scheme ................................. 184 5.4.2 Godunov's Method ............................... 185
5.5 Sample Numerical Results ............................... 189 5.5.1 Linear Advection ................................. 189 5.5.2 The Inviscid Burgers Equation ..................... 191
5.6 FORTRAN 77 Program for Godunov's Method ............. 192
6. The Method of Godunov for Non-linear Systems ......... 201 6.1 Bases of Godunov's Method ............................. 201 6.2 The Godunov Scheme ................................... 204 6.3 Godunov's Method for the Euler Equations ................ 206
6.3.1 Evaluation of the Intercell Fluxes ................... 207 6.3.2 Time Step Size ................................... 209 6.3.3 Boundary Conditions ............................. 210
6.4 Numerical Results and Discussion ........................ 213 6.4.1 Numerical Results for Godunov's Method ........... 214 6.4.2 Numerical Results from Other Methods ............. 217
7. Random Choice and Related Methods .................... 221 7.1 Introduction ........................................... 221 7.2 RCM on a Non-Staggered Grid .......................... 222
7.2.1 The Scheme for Non-Linear Systems ................ 223 7.2.2 Boundary Conditions and the Time Step Size ........ 227
7.3 A Random Choice Scheme of the Lax-Friedrichs Type ...... 228 7.3.1 Review of the Lax-Friedrichs Scheme ............... 228 7.3.2 The Scheme ..................................... 229
7.4 The RCM on a Staggered Grid ........................... 231 7.4.1 The Scheme for Non-Linear Systems ................ 231 7.4.2 A Deterministic First-Order Centred Scheme (FORCE) 231 7.4.3 Analysis of the FORCE Scheme ..................... 233
7.5 Random Numbers ...................................... 234 7.5.1 Van der Corput Pseudo-Random Numbers .......... 234
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7.5.2 Statistical Properties ............................. 235 7.5.3 Propagation of a Single Shock ...................... 237
7.6 Numerical Results ...................................... 239 7.7 Concluding Remarks .................................... 240
8. Flux Vector Splitting Methods ............................ 249 8.1 Introduction ........................................... 249 8.2 The Flux Vector Splitting Approach ...................... 250
8.2.1 Upwind Differencing .............................. 250 8.2.2 The FVS Approach ............................... 252
8.3 FVS for the Isothermal Equations ........................ 254 8.3.1 Split Fluxes ..................................... 254 8.3.2 FVS Numerical Schemes .......................... 256
8.4 FVS Applied to the Euler Equations ...................... 257 8.4.1 Recalling the Equations ........................... 258 8.4.2 The Steger-Warming Splitting ..................... 259 8.4.3 The van Leer Splitting ............................ 260 8.4.4 The Liou-Steffen Scheme .......................... 262
8.5 Numerical Results ...................................... 263 8.5.1 Tests ........................................... 263 8.5.2 Results for Test 1 ................................ 264 8.5.3 Results for Test 2 ................................ 264 8.5.4 Results for Test 3 ................................ 265 8.5.5 Results for Test 4 ................................ 265
9. Approximate-State Riemann Solvers ..................... 273 9.1 Introduction ........................................... 273 9.2 The Riemann Problem and the Godunov Flux ............. 274
9.2.1 Tangential Velocity Components ................... 276 9.2.2 Sonic Rarefactions ................................ 276
9.3 Primitive Variable Riemann Solvers (PVRS) ............... 277 9.4 Approximations Based on the Exact Solver ................ 281
9.4.1 The Two-Rarefaction Riemann Solver (TRRS) ....... 281 9.4.2 A Two-Shock Riemann Solver (TSRS) .............. 283
9.5 Adaptive Riemann Solvers ............................... 284 9.5.1 A PVRS-EXACT Adaptive Scheme ................ 284 9.5.2 A PVRS-TRRS-TSRS Adaptive Scheme ............ 285
9.6 Numerical Results ...................................... 286
10. The HLL and HLLC Riemann Solvers .................... 293 10.1 The Riemann Problem and the Godunov Flux ............. 294 10.2 The Riemann Problem and Integral Relations .............. 295 10.3 The HLL Approximate Riemann Solver ................... 297 10.4 The HLLC Approximate Riemann Solver .................. 299 10.5 Wave-Speed Estimates .................................. 302
Table of Contents XV
10.5.1 Direct Wave Speed Estimates ...................... 302 10.5.2 Pressure-Velocity Based Wave Speed Estimates ...... 303
10.6 Summary of the HLL and HLLC Riemann Solvers .......... 305 10.7 Contact Waves and Passive Scalars ....................... 306 10.8 Numerical Results ...................................... 307 10.9 Closing Remarks and Extensions ......................... 308
11. The Riemann Solver of Roe .............................. 313 11.1 Bases of the Roe Approach .............................. 314
11.1.1 The Exact Riemann Problem and the Godunov Flux .. 314 11.1.2 Approximate Conservation Laws ................... 315 11.1.3 The Approximate Riemann Problem and the Intercell
Flux ............................................ 317 11.2 The Original Roe Method ............................... 319
11.2.1 The Isothermal Equations ......................... 320 11.2.2 The Euler Equations .............................. 322
11.3 The Roe-Pike Method .................................. 326 11.3.1 The Approach ................................... 326 11.3.2 The Isothermal Equations ......................... 327 11.3.3 The Euler Equations .............................. 331
11.4 An Entropy Fix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 11.4.1 The Entropy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 11.4.2 The Harten-Hyman Entropy Fix ................... 335 11.4.3 The Speeds u., a.L, a*R .......................... 337
11.5 Numerical Results and Discussion ........................ 339 11.5.1 The Tests ....................................... 339 11.5.2 The Results ..................................... 340
11.6 Extensions ............................................. 341
12. The Riemann Solver of Osher ............................. 345 12.1 Osher's Scheme for a General System ..................... 346
12.1.1 Mathematical Bases .............................. 346 12.1.2 Osher's Numerical Flux ........................... 348 12.1.3 Osher's Flux for the Single-Wave Case .............. 349 12.1.4 Osher's Flux for the Inviscid Burgers Equation ....... 351 12.1.5 Osher's Flux for the General Case .................. 352
12.2 Osher's Flux for the Isothermal Equations ................. 353 12.2.1 Osher's Flux with P-Ordering ..................... 354 12.2.2 Osher's Flux with 0-0rdering ..................... 357
12.3 Osher's Scheme for the Euler Equations .................. 360 12.3.1 Osher's Flux with P-Ordering ..................... 361 12.3.2 Osher's Flux with 0-0rdering ..................... 363 12.3.3 Remarks on Path Orderings ....................... 368 12.3.4 The Split Three-Dimensional Case ................. 371
12.4 Numerical Results and Discussion ........................ 372
XVI Table of Contents
12.5 Extensions ............................................. 373
13. High-Order and TVD Methods for Scalar Problems ...... 379 13.1 Introduction ........................................... 379 13.2 Basic Properties of Selected Schemes ...................... 381
13.2.1 Selected Schemes ................................. 381 13.2.2 Accuracy ........................................ 383 13.2.3 Stability ........................................ 384
13.3 WAF-Type High Order Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 385 13.3.1 The Basic WAF Scheme ............................ 386 13.3.2 Generalisations of the WAF Scheme ................. 389
13.4 MUSCL-Type High-Order Methods ...................... 391 13.4.1 Data Reconstruction .............................. 392 13.4.2 The MUSCL-Hancock Method (MHM) ............. 394 13.4.3 The Piece-Wise Linear Method (PLM) .............. 397 13.4.4 The Generalised Riemann Problem (GRP) Method ... 399 13.4.5 MUSCL-Hancock Centred (sLic) Schemes ........... 402 13.4.6 Other Approaches ................................ 404 13.4. 7 Semi-Discrete Schemes ............................ 405 13.4.8 Implicit Methods ................................. 405
13.5 Monotone Schemes and Accuracy ......................... 406 13.5.1 Monotone Schemes ............................... 406 13.5.2 A Motivating Example ............................ 409 13.5.3 Monotone Schemes and Godunov's Theorem ......... 413 13.5.4 Spurious Oscillations and High Resolution ........... 414 13.5.5 Data Compatibility ............................... 415
13.6 Total Variation Diminishing (TVD) Methods .............. 417 13.6.1 The Total Variation .............................. 418 13.6.2 TVD and Monotonicity Preserving Schemes ......... 419
13.7 Flux Limiter Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 13. 7.1 TVD Version of the WAF Method ................... 422 13.7.2 The General Flux-Limiter Approach ................ 429 13. 7.3 TVD Upwind Flux Limiter Schemes ................ 435 13.7.4 TVD Centred Flux Limiter Schemes ................ 439
13.8 Slope Limiter Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 13.8.1 TVD Conditions ................................. 446 13.8.2 Construction of TVD Slopes ....................... 447 13.8.3 Slope Limiters ................................... 448 13.8.4 Limited Slopes Obtained from Flux Limiters ......... 450
13.9 Extensions of TVD Methods ............................. 451 13.9.1 TVD Schemes in the Presence of Source Terms ....... 451 13.9.2 TVD Schemes in the Presence of Diffusion Terms ..... 452
13.10Numerical Results for Linear Advection ................... 453
Table of Contents XVII
14. High-Order and TVD Schemes for Non-Linear Systems .. 459 14.1 Introduction ........................................... 459 14.2 CFL and Boundary Conditions ........................... 460 14.3 Weighted Average Flux (WAF) Schemes ................... 462
14.3.1 The Original Version of WAF ....................... 462 14.3.2 A Weighted Average State Version ................. 464 14.3.3 Rarefactions in State Riemann Solvers .............. 465 14.3.4 TVD Version of WAF Schemes ...................... 467 14.3.5 Riemann Solvers ................................. 469 14.3.6 Summary of the WAF Method ...................... 469
14.4 The MUSCL-Hancock Scheme ........................... 470 14.4.1 The Basic Scheme ................................ 470 14.4.2 A Variant of the Scheme .......................... 472 14.4.3 TVD Version of the Scheme ....................... 473 14.4.4 Summary of the MUSCL-Hancock Method .......... 476
14.5 Centred TVD Schemes .................................. 477 14.5.1 Review of the FORCE Flux ......................... 477 14.5.2 A Flux Limiter Centred (FLIC) Scheme .............. 478 14.5.3 A Slope Limiter Centred (sLic) Scheme ............. 480
14.6 Primitive-Variable Schemes .............................. 481 14.6.1 Formulation of the Equations and Primitive Schemes . 481 14.6.2 A WAF-Type Primitive Variable Scheme ............. 482 14.6.3 A MUSCL-Hancock Primitive Scheme .............. 485 14.6.4 Adaptive Primitive-Conservative Schemes ........... 487
14.7 Some Numerical Results ................................. 488 14.7.1 Upwind TVD Methods ............................ 489 14.7.2 Centred TVD Methods ............................ 489
15. Splitting Schemes for PDEs with Source Terms ........... 497 15.1 Introduction ........................................... 497 15.2 Splitting for a Model Equation ........................... 498 15.3 Numerical Methods Based on Splitting .................... 501
15.3.1 Model Equations ................................. 501 15.3.2 Schemes for Systems .............................. 502
15.4 Remarks on ODE Solvers ................................ 503 15.4.1 First-Order Systems of ODEs ...................... 503 15.4.2 Numerical Methods ............................... 505 15.4.3 Implementation Details for Split Schemes ............ 506
15.5 Concluding Remarks .................................... 507
16. Methods for Multi-Dimensional PDEs .................... 509 16.1 Introduction ........................................... 509 16.2 Dimensional Splitting ................................... 510
16.2.1 Splitting for a Model Problem ..................... 510 16.2.2 Splitting Schemes for Two-Dimensional Systems ..... 511
XVIII Table of Contents
16.2.3 Splitting Schemes for Three-Dimensional Systems .... 513 16.3 Practical Implementation of Splitting Schemes in Three Di-
mensions .............................................. 515 16.3.1 Handling the Sweeps by a Single Subroutine ......... 515 16.3.2 Choice of Time Step Size .......................... 517 16.3.3 The Intercell Flux and the TVD Condition ........... 517
16.4 Unsplit Finite Volume Methods .......................... 521 16.4.1 Introductory Concepts ............................ 521 16.4.2 Accuracy and Stability of Multidimensional Schemes .. 524
16.5 A MuscL-Hancock Finite Volume Scheme ................. 527 16.6 WAF-Type Finite Volume Schemes ....................... 529
16.6.1 Two-Dimensional Linear Advection ................. 529 16.6.2 Three-Dimensional Linear Advection ............... 533 16.6.3 Schemes for Two-Dimensional Nonlinear Systems .... 536 16.6.4 Schemes for Three-Dimensional Nonlinear Systems ... 539
16.7 Non-Cartesian Geometries ............................... 540 16.7.1 Introduction ..................................... 540 16. 7.2 General Domains and Coordinate Transformation .... 541 16.7.3 The Finite Volume Method for Non-Cartesian Domains543
17. Multidimensional Test Problems .......................... 551 17.1 Explosions and Implosions ............................... 551
17.1.1 Explosion Test in Two-Space Dimensions ........... 552 17.1.2 Implosions in Two Dimensions ..................... 555 17.1.3 Explosion Test in Three Space Dimensions ........... 556
17.2 Shock Wave Reflection from Wedges ...................... 557 17.2.1 Mach Number Ms = 1.7 and <P = 25 Degrees ......... 558 17.2.2 Mach Number Ms = 1.2 and ¢ = 30 Degrees ......... 561
18. Concluding Remarks ..................................... 563 18.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 18.2 Extensions and Applications ............................. 564
18.2.1 Shallow Water Equations .......................... 564 18.2.2 Steady Supersonic Euler Equations ................. 564 18.2.3 The Artificial Compressibility Equations ............ 565 18.2.4 The Compressible Navier-Stokes Equations .......... 565 18.2.5 Compressible Materials ........................... 565 18.2.6 Detonation Waves ................................ 565 18.2. 7 Multiphase Flows ................................ 566 18.2.8 Magnetohydrodynamics (MHD) .................... 566 18.2.9 Relativistic Gas Dynamics ......................... 566 18.2.10Waves in Solids .................................. 567
18.3 Teaching and Development Programs ..................... 567
