Volume 21, Issue 4
Numerical Study of Partially Conservative Moment Equations in Kinetic Theory

Julian Koellermeier & Manuel Torrilhon

Commun. Comput. Phys., 21 (2017), pp. 981-1011.

Published online: 2018-04

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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 nonconservative 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.

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@Article{CiCP-21-981, author = {Julian Koellermeier and Manuel Torrilhon}, title = {Numerical Study of Partially Conservative Moment Equations in Kinetic Theory}, journal = {Communications in Computational Physics}, year = {2018}, volume = {21}, number = {4}, pages = {981--1011}, 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 nonconservative 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0053}, url = {http://global-sci.org/intro/article_detail/cicp/11268.html} }
TY - JOUR T1 - Numerical Study of Partially Conservative Moment Equations in Kinetic Theory AU - Julian Koellermeier & Manuel Torrilhon JO - Communications in Computational Physics VL - 4 SP - 981 EP - 1011 PY - 2018 DA - 2018/04 SN - 21 DO - http://dor.org/10.4208/cicp.OA-2016-0053 UR - https://global-sci.org/intro/cicp/11268.html KW - AB -

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 nonconservative 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.

Julian Koellermeier & Manuel Torrilhon. (1970). Numerical Study of Partially Conservative Moment Equations in Kinetic Theory. Communications in Computational Physics. 21 (4). 981-1011. doi:10.4208/cicp.OA-2016-0053
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