Numerical approximation of hyperbolic systems of conservation laws
type: Item
instance of: scholarly article
title: Numerical approximation of hyperbolic systems of conservation laws
zbMATH Open document ID: 0860.65075
author: Edwige Godlewski
author: P.-A. Raviart
published in: Applied Mathematical Sciences
publication date: 1996-12-05T00:00:00Z
review text: This work is devoted to the theory and approximation of nonlinear hyperbolic 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 [Hyperbolic systems of conservation laws, Mathématiques et Applications, Ellipses, Paris, (1991; Zbl 0768.35059)] in which the scalar case is covered. The authors 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. This book is divided into six chapters, including an introductory chapter. In this introduction the authors resume the notions that are necessary for a self-sufficient understanding of this book (definitions of hyperbolicity, weak solution, and entropy), present practical examples of systems of conservation laws in several space variables that arise in continuum physics and recall the main results concerning the scalar case. Chapter I is devoted to the resolution of the Riemann problem for a general hyperbolic system in one space dimension, and introduces the classical notions of Riemann invariants and simple waves, the rarefaction and shock curves, and characteristics and entropy conditions. These notions are illustrated on the examples of the \(p\)-systems and the gas dynamics equations. In Chapter II the authors prove that the Riemann problem for the one-dimensional system of gas dynamics is always solvable. These two basic chapters present the results with detailed proofs. In Chapter III the authors develop the most usual numerical scheme for one-dimensional hyperbolic systems with special emphasis on the application to gas dynamics. A short account on the kinetic theory and the kinetic schemes are given. Chapter IV is devoted to the study of finite volume methods for bidimensional systems, preceded by some theoretical considerations on multidimensional systems. An introduction t…
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 76M20
Mathematics Subject Classification ID: 76N15
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 76-02
zbMATH DE Number: 954087
zbMATH Keywords: textbook
zbMATH Keywords: finite difference methods
zbMATH Keywords: nonlinear hyperbolic systems of conservation laws
zbMATH Keywords: Riemann problem
zbMATH Keywords: shock
zbMATH Keywords: gas dynamics
zbMATH Keywords: finite volume methods
zbMATH Keywords: bibliography
reviewed by: Karel Najzar
describes a project that uses: HLLE
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 954087