Godlewski E., Raviart P.-A. Hyperbolic systems of conservation laws

Paris: Ellipse, 1991. — 258 p.The material covered by this publication is an extended version of,a one-semester post-graduate course in Numerical Analysis taught at the University Pierre et Marie Curie and at the Ecole Poly technique.
Nonlinear hyperbolic systems of conservation laws playa central role in Science and Engineering and their mathematical theory as well as their numerical approximation have made recent significative progress, The purpose of these notes is to serve as an introduction to this important topic, For the sake of simplicity and brevity, we have essentially restricted ourselves to scalar conservation laws since the mathematical theory is rather
complete only in this case. Although real-life applications mainly involve systems of conservation laws, many ideas developed for scalar conservation laws provide a useful basis for solving systems,
The book is divided into four chapters, In the first chapter, we consider general systems of conservation laws, After having presented a number of practical examples of such systems, we develop the notions of weak solution and entropy,
Chapter II is devoted to the mathematical theory of the Cauchy problem for scalar conservation laws in several space dimensions, The existence of the entropy solution is obtained by the vanishing viscosity method, The uniqueness follows from Kruzkov's resultwhich states that the entropy solution depends continuously on the initial condition in L 'Ioc' Finally we construct the entropy solution of the Riemann problem in one space dimension,
In Chapter III we begin the study of finite difference approximations of one-dimensional scalar conservation laws, We introduce the basic notions concerning explicit difference schemes and particularly the T,V,D, property which implies the convergence of these schemes and plays an important role in the design of efficient numerical methods, Then we construct the main classes of first order schemes,
In most practical problems, first order schemes are not sufficiently accurate so that we have to use second order difference schemes, Chapter IV is in fact devoted to the construction of T,V,D, second order schemes. We present essentially two general procedures; the antidiffusion method and Van Leer's method which give rise to most of accurate stable sche'mes and can be generalized in a natural way to systems of conservation laws,