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On the connection between some Riemann-solver free approachesto the approximation of multi-dimensional systems of hyperbolicconservation laws

Published online by Cambridge University Press:  15 December 2004

Tim Kröger ,
Sebastian Noelle  and
Susanne Zimmermann
Tim Kröger
Affiliation:
RWTH Aachen, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056 Aachen, Germany. kroeger@igpm.rwth-aachen.de.; noelle@igpm.rwth-aachen.de.
Sebastian Noelle
Affiliation:
RWTH Aachen, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056 Aachen, Germany. kroeger@igpm.rwth-aachen.de.; noelle@igpm.rwth-aachen.de.
Susanne Zimmermann
Affiliation:
ETH Zentrum, Seminar für Angewandte Mathematik, 8092 Zürich, Switzerland.
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Abstract

In this paper, we present some interesting connections between anumber of Riemann-solver free approaches to the numerical solutionof multi-dimensional systems of conservation laws. As a main part,we present a new and elementary derivation of Fey's Method ofTransport (MoT) (respectively the second author's ICE version ofthe scheme) and the state decompositions which form the basis of it.The only tools that we use are quadrature rules applied to themoment integral used in the gas kinetic derivation of the Eulerequations from the Boltzmann equation, to the integration in timealong characteristics and to space integrals occurring in the finitevolume formulation. Thus, we establish a connection between theMoT approach and the kinetic approach. Furthermore,Ostkamp's equivalence result between her evolution Galerkin schemeand the method of transport is lifted up from the level ofdiscretizations to the level of exact evolution operators,introducing a new connection between the MoT and theevolution Galerkin approach. At the same time, we clarifysome important differences between these two approaches.

Keywords

Systems of conservation lawsFey's method of transportEuler equationsBoltzmann equationkinetic schemesbicharacteristic theorystate decompositionsflux decompositionsexact and approximate integral representationsquadrature rules.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 6 , November 2004 , pp. 989 - 1009
Copyright
© EDP Sciences, SMAI, 2004

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