A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling

Anaïs Crestetto (1, 2, 3) , Nicolas Crouseilles (4, 5, 3, 2) , Mohammed Lemou (6, 4, 5, 3, 2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 MINGUS - Multi-scale numerical geometric schemes
3 Inria Rennes – Bretagne Atlantique
4 IRMAR - Institut de Recherche Mathématique de Rennes
5 UR - Université de Rennes
6 CNRS - Centre National de la Recherche Scientifique

In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part has to be performed, and the stiffness of both these two parts prevent from uniform stability. To overcome this difficulty, the micro-macro system is reformulated into a continuous PDE whose coefficients are no longer stiff, and depend on the time step ∆t in a consistent way. This non-stiff reformulation of the micro-macro system allows the use of standard particle approximations for the transport part, and extends the work in [Crestetto, Crouseilles, Lemou, KRM, 2012] where a particle approximation has been applied using a micro-macro decomposition on kinetic equations in the fluid scaling. Beyond the so-called asymptotic-preserving property which is satisfied by our schemes, they significantly reduce the inherent noise of traditional particle methods, and they have a computational cost which decreases as the system approaches the diffusion limit.

Domaines: Mathématiques [math]; Analyse numérique [math.NA]

Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling. Communications in Mathematical Sciences, 2018, 16 (4), pp.887-911. ⟨10.4310/CMS.2018.v16.n4.a1⟩. ⟨hal-01439288v2⟩