Numerical Approximation of Hyperbolic Systems of Conservation Laws, Issue 118

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Springer Science & Business Media, 1996 - Computers - 509 pages
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This work is devoted to the theory and approximation of nonlinear hyper bolic systems of conservation laws in one or two space variables. It follows directly a previous publication on hyperbolic systems of conservation laws by the same authors, and we shall make frequent references to Godlewski and Raviart (1991) (hereafter noted G. R. ), though the present volume can be read independently. This earlier publication, apart from a first chap ter, especially covered the scalar case. Thus, we shall detail here neither the mathematical theory of multidimensional scalar conservation laws nor their approximation in the one-dimensional case by finite-difference con servative schemes, both of which were treated in G. R. , but we shall mostly consider systems. The theory for systems is in fact much more difficult and not at all completed. This explains why we shall mainly concentrate on some theoretical aspects that are needed in the applications, such as the solution of the Riemann problem, with occasional insights into more sophisticated problems. The present book is divided into six chapters, including an introductory chapter. For the reader's convenience, we shall resume in this Introduction the notions that are necessary for a self-sufficient understanding of this book -the main definitions of hyperbolicity, weak solutions, and entropy present the practical examples that will be thoroughly developed in the following chapters, and recall the main results concerning the scalar case.
 

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Contents

Introduction
1
2 Weak solutions of systems of conservation laws
11
3 Entropy solutions
21
Notes
35
Nonlinear hyperbolic systems in one space dimension
37
2 The nonlinear case Definitions and examples
40
3 Simple waves and Riemann invariants
49
4 Shock waves and contact discontinuities
60
4 The Osher scheme
229
5 Flux vector splitting methods
237
6 Van Leers secondorder method
245
7 Kinetic schemes for the Euler equations
269
Notes
301
The case of multidimensional systems
303
2 The gas dynamics equations in two space dimensions
316
3 Multidimensional finite difference schemes
343

5 Characteristic curves and entropy conditions
70
6 Solution of the Riemann problem
83
7 The Riemann problem for the psystem
87
Notes
97
Gas dynamics and reacting flows
99
2 Entropy satisfying shock conditions
108
3 Solution of the Riemann problem
126
4 Reacting flows The ChapmanJouguet theory
142
5 Reacting flows The ZND model for detonations
160
Notes
166
Finite difference schemes for onedimensional systems
167
2 Godunovs method
182
3 Roes method
196
4 Finitevolume methods
360
5 Secondorder finitevolume schemes
403
Notes
415
An introduction to boundary conditions
417
2 The nonlinear approach
435
3 Gas dynamics
442
4 Absorbing boundary conditions
446
5 Numerical treatment
453
Notes
460
Bibliography
461
Index
501
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