A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit

Mohammed Lemou (1) , Luc Mieussens (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 IMB - Institut de Mathématiques de Bordeaux

We propose a new numerical scheme for linear transport equations. It is based on a decomposition of the distribution function into equilibrium and nonequilibrium parts. We also use a projection technique that allows us to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the nonequilibrium part. By using a suitable time semi-implicit discretization, our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes. It is asymptotic preserving in the following sense: when the mean free path of the particles is small, our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model. A uniform stability property is proved for the simple telegraph model. Various boundary conditions are studied. Our method is validated in one-dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes.

Mots clés: transport equations; diffusion limit; asymptotic preserving schemes; stiff terms

Domaines: Analyse numérique [math.NA]

Mohammed Lemou, Luc Mieussens. A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit. SIAM Journal on Scientific Computing, 2008, 31 (1), pp.334-368. ⟨10.1137/07069479X⟩. ⟨hal-00348594⟩