Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit

Nicolas Crouseilles (1, 2) , Hélène Hivert (1, 2) , Mohammed Lemou (1, 2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 IPSO - Invariant Preserving SOlvers

We construct numerical schemes to solve kinetic equations with anomalous diusion scaling. When the equilib-rium is heavy-tailed or when the collision frequency degenerates for small velocities, an appropriate scaling shouldbe made and the limit model is the so-called anomalous or fractional diusion model. Our first scheme is based ona suitable micro-macro decomposition of the distribution function whereas our second scheme relies on a Duhamelformulation of the kinetic equation. Both are Asymptotic Preserving (AP): they are consistent with the kineticequation for all fixed value of the scaling parameter epsilon> 0 and degenerate into a consistent scheme solving theasymptotic model when epsilon tends to 0. The second scheme enjoys the stronger property of being uniformly accurate(UA) with respect to epsilon. The usual AP schemes known for the classical diusion limit cannot be directly appliedto the context of anomalous diusion scaling, since they are not able to capture the important eects of largeand small velocities. We present numerical tests to highlight the eciency of our schemes.

Mots clés: anomalous diffusion; kinetic model; numerical scheme; Asymptotic Preserving scheme

Domaines: Analyse numérique [math.NA]

Nicolas Crouseilles, Hélène Hivert, Mohammed Lemou. Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit. Comptes Rendus. Mathématique, 2015, 353 (8), pp.755-760. ⟨10.1016/j.crma.2015.05.003⟩. ⟨hal-01151518⟩