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Numerical Study of Partially Conservative Moment Equations in Kinetic Theory

  • Julian Koellermeier (a1) and Manuel Torrilhon (a1)


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.


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*Corresponding author. Email addresses: (J. Koellermeier), (M. Torrilhon)


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[1] Au, J. D., Torrilhon, M., and Weiss, W.. The shocktube-experiment in extended thermodynamics. Phys. Fluids, 13(8):24232432, 2001.
[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):511525, 1954.
[3] Bird, G. A.. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford: Clarendon Press, 1994.
[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):547571, 2013.
[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., 33:291303, 2010.
[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):834844, 2009.
[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):457486, 2016.
[8] Grad, H.. On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2(4):331407, 1949.
[9] Kauf, P.. Multi-Scale Approximation Models for the Boltzmann Equation. PhD thesis, ETH Zrich, 2011.
[10] Koellermeier, J.. Hyperbolic approximation of kinetic equations using quadrature-based projection methods. Master's thesis, RWTH Aachen University, 2013.
[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):531549, 2014.
[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.
[13] LeVeque, R. J.. Wave propagation algorithms for multidimensional hyperbolic systems. J. Comp. Phys., 131:327353, 1997.
[14] LeVeque, R.J.. Finite Volume Methods for Hyperbolic Problems. Cambridge, 2002.
[15] Levermore, C. D.. Moment closure hierarchies for kinetic theories. J. Stat. Phys., 83(5–6):10211065, 1996.
[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):821852, 2004.
[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., 227:18871922, 2008.
[18] Schärer, R.P. and Torrilhon, M.. On singular closures for the 5-moment system in kinetic gas theory. Commun. Comput. Phys., 17(2):371400, 2015.
[19] Stecca, G.. Numerical modelling of gravel-bed river morphodynamics. PhD thesis, Universita’ Degli Studi di Trento, 2012.
[20] Struchtrup, H.. Macroscopic Transport Equations for Rarefied Gas Flows. Springer, 2005.
[21] Toro, E. F. and Billett, S. J.. Centred tvd schemes for hyperbolic conservation laws. IMA J. Num. Anal., 20:4779, 2000.
[22] Torrilhon, M.. Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions. Commun. Comput. Phys., 7(4):639673, 2010.
[23] White, F. M.. Fluid Mechanics. WCB/McGraw-Hill, 1999.
[24] Yong, W.-A.. Singular perturbations of first-order hyperbolic systems with stiff source terms. J. Diff. Equations, 155:89132, 1999.


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Numerical Study of Partially Conservative Moment Equations in Kinetic Theory

  • Julian Koellermeier (a1) and Manuel Torrilhon (a1)


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