Numerical Approximation of Hyperbolic Systems of Conservation Laws, Issue 118
Springer Science & Business Media, 1996 - Computers - 509 pages
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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2 Weak solutions of systems of conservation laws
3 Entropy solutions
Nonlinear hyperbolic systems in one space dimension
2 The nonlinear case Definitions and examples
3 Simple waves and Riemann invariants
4 Shock waves and contact discontinuities
4 The Osher scheme
5 Flux vector splitting methods
6 Van Leers secondorder method
7 Kinetic schemes for the Euler equations
The case of multidimensional systems
2 The gas dynamics equations in two space dimensions
3 Multidimensional finite difference schemes
5 Characteristic curves and entropy conditions
6 Solution of the Riemann problem
7 The Riemann problem for the psystem
Gas dynamics and reacting flows
2 Entropy satisfying shock conditions
3 Solution of the Riemann problem
4 Reacting flows The ChapmanJouguet theory
5 Reacting flows The ZND model for detonations
Finite difference schemes for onedimensional systems
2 Godunovs method
3 Roes method
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1-rarefaction 1-shock Afc(u approximation assume boundary conditions cell Chapter characteristic field coefficients compute conservation laws consider constant contact discontinuity corresponding Crussard curve defined deflagration denote derivatives detonation difference scheme differential eigenvalues eigenvectors entropy entropy condition entropy solution equivalently Euler equations Euler system Eulerian coordinates Example exists fcth characteristic field Figure flow formula function gas dynamics equations genuinely nonlinear given gives Godunov's grid Hence Hugoniot curve hyperbolic systems ideal gas INRIA instance integral curve intersection introduce Jacobian matrix jump condition Lagrangian coordinates Lemma linear linearly degenerate matrix method Moreover notations Note numerical flux obtain one-dimensional p-system piecewise Proof rarefaction rarefaction wave Rayleigh line Remark resp result Riemann problem Roe's scheme satisfies scalar Section shock curves smooth solve strictly convex systems of conservation Theorem upwind values variables vector velocity viscosity weak solution yields
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