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

Mounir Bennoune (1) , Mohammed Lemou (2) , Luc Mieussens (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 IMB - Institut de Mathématiques de Bordeaux

In this paper we propose a numerical scheme to solve the Kac model of the Boltzmann equation for multiscale rarefied gas dynamics. This scheme is 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.

Mots clés: kinetic-equations; numerical schemes; relaxation schemes; gas

Domaines: Analyse numérique [math.NA]

Mounir Bennoune, Mohammed Lemou, Luc Mieussens. An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit. Continuum Mechanics and Thermodynamics, 2009, 21 (5), pp.401-421. ⟨10.1007/s00161-009-0116-2⟩. ⟨hal-00383271⟩