Numerical schemes for kinetic equations in the anomalous diffusion limit. Part II: degenerate collision frequency.

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
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
Abstract : In this work, which is the continuation of [9], we propose numerical schemes for linear kinetic equation which are able to deal with the fractional diffusion limit. When the collision frequency degenerates for small velocities it is known that for an appropriate time scale, the small mean free path limit leads to an anomalous diffusion equation. From a numerical point of view, this degeneracy gives rise to an additional stiffness that must be treated in a suitable way to avoid a prohibitive computational cost. Our aim is therefore to construct a class of numerical schemes which are able to undertake these stiffness. This means that the numerical schemes are able to capture the effect of small velocities in the small mean free path limit with a fixed set of numerical parameters. Various numerical tests are performed to illustrate the efficiency of our methods in this context.
Keywords : Fractional diffusion equation Asymptotic preserving schemes BGK equation Anomalous diffusion limit Degenerate collision frequency
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SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2016, 38 (4)
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Nicolas Crouseilles, Hélène Hivert, Mohammed Lemou. Numerical schemes for kinetic equations in the anomalous diffusion limit. Part II: degenerate collision frequency. . SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2016, 38 (4). 〈hal-01245312〉

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