Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-19T02:58:57.440Z Has data issue: false hasContentIssue false
  • English
  • Français

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:[email protected] (J. Koellermeier), [email protected] (M. Torrilhon)
*Corresponding author. Email addresses:[email protected] (J. Koellermeier), [email protected] (M. Torrilhon)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Au, J. D., Torrilhon, M., and Weiss, W.. The shocktube-experiment in extended thermodynamics. Phys. Fluids, 13(8):2423–2432, 2001.CrossRefGoogle Scholar
[2] Bhatnagar, P. L., Gross, E. P., and Krook, M.. A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94(3):511–525, 1954.Google Scholar
[3] Bird, G. A.. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford: Clarendon Press, 1994.Google Scholar
[4] Cai, Z., Fan, Y., and Li, R.. Globally hyperbolic regularization of Grad's moment system in one dimensional space. Comm. Math. Sci., 11(2):547–571, 2013.Google Scholar
[5] Canestrelli, A., Dumbser, M., Siviglia, A., and Toro, E.F.. Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed. Adv. Water Resour., 33:291–303, 2010.Google Scholar
[6] Canestrelli, A., Siviglia, A., Dumbser, M., and Toro, E.F.. Well-balanced high-order centred schemes for non-conservative hyperbolic systems. applications to shallow water equations with fixed and mobile bed. Adv. Water Resour., 32(6):834–844, 2009.CrossRefGoogle Scholar
[7] Fan, Y., Koellermeier, J., Li, J., Li, R., and Torrilhon, M.. Model reduction of kinetic equations by operator projection. J. Stat. Phys., 162(2):457–486, 2016.Google Scholar
[8] Grad, H.. On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2(4):331–407, 1949.Google Scholar
[9] Kauf, P.. Multi-Scale Approximation Models for the Boltzmann Equation. PhD thesis, ETH Zrich, 2011.Google Scholar
[10] Koellermeier, J.. Hyperbolic approximation of kinetic equations using quadrature-based projection methods. Master's thesis, RWTH Aachen University, 2013.CrossRefGoogle Scholar
[11] Koellermeier, J., Schaerer, R., and Torrilhon, M.. A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods. Kinet. Relat. Mod., 7(3):531–549, 2014.Google Scholar
[12] Koellermeier, J. and Torrilhon, M.. On new hyperbolic moment models for the boltzmann equation. In Conference Proceedings of the YIC GACM 2015, Publication Server of RWTH Aachen University, 2015.Google Scholar
[13] LeVeque, R. J.. Wave propagation algorithms for multidimensional hyperbolic systems. J. Comp. Phys., 131:327–353, 1997.CrossRefGoogle Scholar
[14] LeVeque, R.J.. Finite Volume Methods for Hyperbolic Problems. Cambridge, 2002.Google Scholar
[15] Levermore, C. D.. Moment closure hierarchies for kinetic theories. J. Stat. Phys., 83(5–6):1021–1065, 1996.Google Scholar
[16] Pars, C. and Castro, M.. On the well-balance property of roe's method for nonconservative hyperbolic systems. applications to shallow-water systems. Math. Model. Anal., 38(5):821–852, 2004.Google Scholar
[17] Rhebergen, S., Bokhove, O., and van der Vegt, J.J.W.. Discontinuous galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comp. Phys., 227:1887–1922, 2008.Google Scholar
[18] Schärer, R.P. and Torrilhon, M.. On singular closures for the 5-moment system in kinetic gas theory. Commun. Comput. Phys., 17(2):371–400, 2015.Google Scholar
[19] Stecca, G.. Numerical modelling of gravel-bed river morphodynamics. PhD thesis, Universita’ Degli Studi di Trento, 2012.Google Scholar
[20] Struchtrup, H.. Macroscopic Transport Equations for Rarefied Gas Flows. Springer, 2005.Google Scholar
[21] Toro, E. F. and Billett, S. J.. Centred tvd schemes for hyperbolic conservation laws. IMA J. Num. Anal., 20:47–79, 2000.Google Scholar
[22] Torrilhon, M.. Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions. Commun. Comput. Phys., 7(4):639–673, 2010.Google Scholar
[23] White, F. M.. Fluid Mechanics. WCB/McGraw-Hill, 1999.Google Scholar
[24] Yong, W.-A.. Singular perturbations of first-order hyperbolic systems with stiff source terms. J. Diff. Equations, 155:89–132, 1999.CrossRefGoogle Scholar