On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws

Tim Kröger; Sebastian Noelle; Susanne Zimmermann[1]

  • [1] ETH Zentrum, Seminar für Angewandte Mathematik, 8092 Zürich, Switzerland.

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 6, page 989-1009
  • ISSN: 0764-583X

Abstract

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In this paper, we present some interesting connections between a number of Riemann-solver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey’s Method of Transport (MoT) (respectively the second author’s ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation of the Euler equations from the Boltzmann equation, to the integration in time along characteristics and to space integrals occurring in the finite volume formulation. Thus, we establish a connection between the MoT approach and the kinetic approach. Furthermore, Ostkamp’s equivalence result between her evolution Galerkin scheme and the method of transport is lifted up from the level of discretizations to the level of exact evolution operators, introducing a new connection between the MoT and the evolution Galerkin approach. At the same time, we clarify some important differences between these two approaches.

How to cite

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Kröger, Tim, Noelle, Sebastian, and Zimmermann, Susanne. "On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.6 (2004): 989-1009. <http://eudml.org/doc/246051>.

@article{Kröger2004,
abstract = {In this paper, we present some interesting connections between a number of Riemann-solver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey’s Method of Transport (MoT) (respectively the second author’s ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation of the Euler equations from the Boltzmann equation, to the integration in time along characteristics and to space integrals occurring in the finite volume formulation. Thus, we establish a connection between the MoT approach and the kinetic approach. Furthermore, Ostkamp’s equivalence result between her evolution Galerkin scheme and the method of transport is lifted up from the level of discretizations to the level of exact evolution operators, introducing a new connection between the MoT and the evolution Galerkin approach. At the same time, we clarify some important differences between these two approaches.},
affiliation = {ETH Zentrum, Seminar für Angewandte Mathematik, 8092 Zürich, Switzerland.},
author = {Kröger, Tim, Noelle, Sebastian, Zimmermann, Susanne},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {systems of conservation laws; Fey’s method of transport; Euler equations; Boltzmann equation; kinetic schemes; bicharacteristic theory; state decompositions; flux decompositions; exact and approximate integral representations; quadrature rules; Galerkin method; method of transport},
language = {eng},
number = {6},
pages = {989-1009},
publisher = {EDP-Sciences},
title = {On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws},
url = {http://eudml.org/doc/246051},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Kröger, Tim
AU - Noelle, Sebastian
AU - Zimmermann, Susanne
TI - On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 6
SP - 989
EP - 1009
AB - In this paper, we present some interesting connections between a number of Riemann-solver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey’s Method of Transport (MoT) (respectively the second author’s ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation of the Euler equations from the Boltzmann equation, to the integration in time along characteristics and to space integrals occurring in the finite volume formulation. Thus, we establish a connection between the MoT approach and the kinetic approach. Furthermore, Ostkamp’s equivalence result between her evolution Galerkin scheme and the method of transport is lifted up from the level of discretizations to the level of exact evolution operators, introducing a new connection between the MoT and the evolution Galerkin approach. At the same time, we clarify some important differences between these two approaches.
LA - eng
KW - systems of conservation laws; Fey’s method of transport; Euler equations; Boltzmann equation; kinetic schemes; bicharacteristic theory; state decompositions; flux decompositions; exact and approximate integral representations; quadrature rules; Galerkin method; method of transport
UR - http://eudml.org/doc/246051
ER -

References

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