Abstract:
A class of numerical schemes is presented to efficiently solve kinetic equations in the anomalous diffusion scaling. Such scaling is relevant for many models in kinetic theory and essentially appears in two important situations: the case of heavy-tailed equilibria and the case of singular collision frequency. Standard Asymptotic Preserving (AP) schemes for the diffusion limit do not work in the case of anomalous diffusion scaling since they are not able to undertake the effect of (too) large or (too) small velocities. We present three different numerical schemes in this case which are all AP: a modified time-implicit scheme, a time-explicit micro/macro numerical scheme and a Duhamel formulation based scheme. The last strategy enjoys the stronger property of being uniformly accurate with respect to the scaling parameter. Numerical tests will be presented to show the efficiency of these strategies. |