Numerical Approximation of Hyperbolic Systems of Conservation Laws

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.