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An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit

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Abstract

In this article, we propose a numerical scheme to solve the Kac model of the Boltzmann equation for multiscale rarefied gas dynamics. Formally, this scheme is shown to be uniformly stable with respect to the Knudsen number, consistent with the fluid-diffusion limit for small Knudsen numbers, and with the Kac equation in the kinetic regime. Our approach is based on the micro–macro decomposition which leads to an equivalent formulation of the Kac model that couples a kinetic equation with macroscopic ones. This method is validated with various test cases and compared to other standard methods.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse (UMR 5219), Université Paul Sabatier, 118, Route de Narbonne, 31062, Toulouse cedex 9, France

    Mounir Bennoune

  2. Institut de Recherche Mathématique de Rennes (IRMAR), (UMR 6625), Université Rennes I, Campus de Beaulieu, 35042, Rennes cedex, France

    Mohammed Lemou

  3. Institut de Mathématiques de Bordeaux (UMR 5251), Université de Bordeaux, 351, cours de la Libération, 33405, Talence cedex, France

    Luc Mieussens

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  1. Mounir Bennoune
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  2. Mohammed Lemou
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  3. Luc Mieussens
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Correspondence to Mounir Bennoune.

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Bennoune, M., Lemou, M. & Mieussens, L. An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit. Continuum Mech. Thermodyn. 21, 401–421 (2009). https://doi.org/10.1007/s00161-009-0116-2

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  • DOI: https://doi.org/10.1007/s00161-009-0116-2

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