Numerical schemes for conservation laws. (English) Zbl 0872.76001

Kröner, Dietmar

Wiley-Teubner Series Advances in Numerical Mathematics. Chichester: Wiley. Stuttgart: Teubner. viii, 508 p. (1997).

We present numerical methods for solving initial value as well as initial-boundary value problems for scalar conservation laws and systems of conservation laws in many dimensions. In particular, modern developments concerning higher order upwind finite volume schemes on unstructed grids and a priori error estimates are taken into account. The details of the mathematical proofs concerning the most important results are included. Theoretical results are presented as far as necessary for the numerical treatment.
The arrangement of the book is as follows. In the introduction we point out the possibility of discontinuous solutions for a simple conservation law in one space dimension, and we present several examples for the occurrence of conservation laws. In chapters 2 and 3 we study the convergence to the entropy solution for finite difference and finite volume schemes for first and higher order on structured and unstructured grids as well as for the streamline diffusion method.
Systems of conservation laws are considered in chapters 4 and 5. At the beginning of chapter 4, we discuss some basic results for systems in one space dimension; we describe the basic ideas of the Glimm scheme and formulate the convergence result for it. Since there are no further theoretical results concerning convergence of numerical schemes for systems in one and several dimensions, we can only describe the most important algorithms. The final chapter 7 is devoted to initial-boundary value problems for conservation laws and to convection-dominated diffusion problems.
This book is a result of a one-year course for students in their third year in applied mathematics and physics held at the universities of Saarbrücken, Heidelberg, Bonn and Freiburg. It can be also recommended for students in engineering science, astrophysics and meteorology or other fields related to this subject.

MSC:

76-02	Research exposition (monographs, survey articles) pertaining to fluid mechanics
76M25	Other numerical methods (fluid mechanics) (MSC2010)
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
35L65	Hyperbolic conservation laws
76M20	Finite difference methods applied to problems in fluid mechanics

Keywords:

initial-boundary value problem; higher order upwind finite volume schemes; unstructured grids; a priori error estimates; discontinuous solutions; convergence; entropy solution; streamline diffusion method; Glimm scheme; convection-dominated diffusion problems