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Inhaltsverzeichnis

Frontmatter

An Introduction to Kinetic Schemes for Gas Dynamics

Abstract
In these notes we present an introduction to the theory of kinetic schemes for gas dynamics. Several ideas developed in recent years on stability and entropy analysis are reviewed. This compendium is intended to be self-contained and self-consistent even though some results are not entirely proved and a preliminary knowledge of elementary hyperbolic theory is supposed (cf Serre [28], Smoller [29], Lax [15]).
Several subjects are not treated: the modifications proposed by Prendergast & Xu [22], [31] and Deshpande [7] for improving accuracy, the early works by Brenier [1] and Giga & Miyakawa [9] on scalar equations, those of Kaniel [12] on gas dynamics with entropy conservation, discrete velocity schemes and the related subject of relaxation schemes.
Benoit Perthame

An Introduction to Nonclassical Shocks of Systems of Conservation Laws

Abstract
We review a recent activity on nonclassical shock waves of strictly hyperbolic systems of conservation laws, generated by balanced diffusion and dispersion effects. These shocks do not satisfy the standard Lax and Liu entropy criteria, and in fact are undercompressive and satisfy a single entropy inequality. The selection of admissible nonclassical shocks requires a strengthened version of the entropy inequality, called a kinetic relation, which constrains the entropy dissipation. The kinetic function is determined from traveling wave solutions to a system of equations augmented with diffusion and dispersion.
For nonconvex scalar conservation laws and non-genuinely nonlinear, strictly hyperbolic systems, the existence and uniqueness of nonclassical shocks is investigated using successively the traveling wave analysis, the front tracking algorithm and the compensated compactness method. Nonclassical shocks may also be generated by finite difference schemes.
The kinetic relation provides a useful tool to study the properties of nonclassical shocks and, in particular, their sensitivity to regularization parameters.
Philippe G. LeFloch

Viscosity and Relaxation Approximation for Hyperbolic Systems of Conservation Laws

Abstract
These lecture notes deal with the approximation of conservation laws via viscosity or relaxation. The following topics are covered:
The general structure of viscosity and relaxation approximations is discussed, as suggested by the second law of thermodynamics, in its form of the Clausius-Duhem inequality. This is done by reviewing models of one dimensional thermoviscoelastic materials, for the case of viscous approximations, and thermomechanical theories with internal variables, for the case of relaxation.
The method of self-similar zero viscosity limits is an approach for constructing solutions to the Riemann problem, as zero-viscosity limits of an elliptic regularization of the Riemann operator. We present recent results on obtaining uniform BV estimates, in a context of strictly hyperbolic systems for Riemann data that are sufficiently close. The structure of the emerging solution, and the connection with shock admissibility criteria is discussed.
The problem of constructing entropy weak solutions for hyperbolic conservation laws via relaxation approximations is considered. We discuss compactness and convergence issues for relaxation approximations converging to the scalar conservation law, in a BV framework, and to the equations of isothermal elastodynamics, via compensated compactness.
Athanasios E. Tzavaras

A Posteriori Error Analysis and Adaptivity for Finite Element Approximations of Hyperbolic Problems

Abstract
The aim of this lecture series is to present an overview of recent developments in the area of a posteriori error estimation for finite element approximations of hyperbolic problems. The approach pursued here rests on the systematic use of hyperbolic duality arguments. We also discuss the question of computational implementation of the a posteriori error bounds into adaptive finite element algorithms.
Endre Süli

Numerical Methods for Gasdynamic Systems on Unstructured Meshes

Abstract
This article considers stabilized finite element and finite volume discretization techniques for systems of conservation laws. Using newly developed techniques in entropy symmetrization theory, simplified forms of the Galerkin least-squares (GLS) and the discontinuous Galerkin (DG) finite element method are developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE system. Detailed consideration is given to symmetrization of the Euler, Navier-Stokes, and magneto-hydrodynamic (MHD) equations. Numerous calculations are presented to evaluate the spatial accuracy and feature resolution capability of the simplified DG and GLS discretizations. Next, upwind finite volume methods are reviewed. Specifically considered are generalizations of Godunov’s method to high order accuracy and unstructured meshes. An important component of high order accurate Godunov methods is the spatial reconstruction operator. A number of reconstruction operators are reviewed based on Green-Gauss formulas as well as least-squares approximation. Several theoretical results using maximum principle analysis are presented for the upwind finite volume method. To assess the performance of the upwind finite volume technique, various numerical calculations in computational fluid dynamics are provided.
Timothy J. Barth

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