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Numerical Study of Partially Conservative Moment Equations in Kinetic Theory

Part of: Hyperbolic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  08 March 2017

Julian Koellermeier  and
Manuel Torrilhon
Julian Koellermeier*
Affiliation:
Department of Mathematics, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany
Manuel Torrilhon*
Affiliation:
Department of Mathematics, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany
*
*Corresponding author. Email addresses:koellermeier@mathcces.rwth-aachen.de (J. Koellermeier), mt@mathcces.rwth-aachen.de (M. Torrilhon)
*Corresponding author. Email addresses:koellermeier@mathcces.rwth-aachen.de (J. Koellermeier), mt@mathcces.rwth-aachen.de (M. Torrilhon)
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Abstract

Moment models are often used for the solution of kinetic equations such as the Boltzmann equation. Unfortunately, standard models like Grad's equations are not hyperbolic and can lead to nonphysical solutions. Newly derived moment models like the Hyperbolic Moment Equations and the Quadrature-Based Moment Equations yield globally hyperbolic equations but are given in partially conservative form that cannot be written as a conservative system.

In this paper we investigate the applicability of different dedicated numerical schemes to solve the partially conservative model equations. Caused by the non-conservative type of equation we obtain differences in the numerical solutions, but due to the structure of the moment systems we show that these effects are very small for standard simulation cases. After successful identification of useful numerical settings we show a convergence study for a shock tube problem and compare the results to a discrete velocity solution. The results are in good agreement with the reference solution and we see convergence considering an increasing number of moments.

Keywords

Kinetic theorymoment methodhyperbolicitynon-conservative numerics

MSC classification

Secondary: 35Q20: Boltzmann equations 35Q35: PDEs in connection with fluid mechanics 35L02: First-order hyperbolic equations
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 4 , April 2017 , pp. 981 - 1011
Copyright
Copyright © Global-Science Press 2017 

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