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An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltzmann Equation

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Abstract

In this paper, we study a time discrete scheme for the initial value problem of the ES-BGK kinetic equation. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We study an implicit-explicit (IMEX) time discretization in which the convection is explicit while the relaxation term is implicit to overcome the stiffness. We first show how the implicit relaxation can be solved explicitly, and then prove asymptotically that this time discretization drives the density distribution toward the local Maxwellian when the mean free time goes to zero while the numerical time step is held fixed. This naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver for the implicit relaxation term. Moreover, it can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. We also show that it is consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. Several numerical examples, in both one and two space dimensions, are used to demonstrate the desired behavior of this scheme.

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Authors and Affiliations

  1. Université de Lyon, CNRS, UMR 5208, Institut Camille Jordan, Université Lyon I, 43, Boulevard du 11 Novembre 1918, 69622, Villeurbanne Cedex, France

    Francis Filbet

  2. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Shi Jin

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  1. Francis Filbet
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  2. Shi Jin
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Correspondence to Francis Filbet.

Additional information

F. Filbet is partially supported by the European Research Council ERC Starting Grant 2009, project 239983-NuSiKiMo. S. Jin was partially supported by NSF grant No. DMS-0608720, NSF FRG grant DMS-0757285, and a Van Vleck Distinguished Research Prize from University of Wisconsin-Madison.

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Filbet, F., Jin, S. An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltzmann Equation. J Sci Comput 46, 204–224 (2011). https://doi.org/10.1007/s10915-010-9394-x

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  • DOI: https://doi.org/10.1007/s10915-010-9394-x

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