Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles

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

1 IRMA - Institut de Recherche Mathématique Avancée
2 CALVI - Scientific computation and visualization
3 IPSO - Invariant Preserving SOlvers
4 IRMAR - Institut de Recherche Mathématique de Rennes

This work is devoted to the numerical simulation of the Vlasov equation in the fluid limit using particles. To that purpose, we first perform a micro-macro decomposition as in \cite{benoune} where asymptotic preserving schemes have been derived in the fluid limit. In \cite{benoune}, a uniform grid was used to approximate both the micro and the macro part of the full distribution function. Here, we modify this approach by using a particle approximation for the kinetic (micro) part, the fluid (macro) part being always discretized by standard finite volume schemes. There are many advantages in doing so: $(i)$ the so-obtained scheme presents a much less level of noise compared to the standard particle method; $(ii)$ the computational cost of the micro-macro model is reduced in the fluid regime since a small number of particles is needed for the micro part; $(iii)$ the scheme is asymptotic preserving in the sense that it is consistent with the kinetic equation in the rarefied regime and it degenerates into a uniformly (with respect to the Knudsen number) consistent (and deterministic) approximation of the limiting equation in the fluid regime.

Mots clés: Vlasov equation; BGK equation; Euler equation; Chapman-Enskog expansion; asymptotic preserving schemes; micro-macro decomposition; particles method

Domaines: Physique Numérique [physics.comp-ph]; Analyse numérique [math.NA]

Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles. Kinetic and Related Models , 2012, 5 (4), pp.787-816. ⟨10.3934/krm.2012.5.787⟩. ⟨hal-00728875⟩