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In this work, we propose some numerical schemes for linear kinetic equationsin the diffusion and anomalous diffusion limit. When the equilibriumdistribution function is a Maxwellian distribution, it is well known that foran appropriate time scale, the small mean free path limit gives rise to adiffusion type equation. However, when a heavy-tailed distribution isconsidered, another time scale is required and the small mean free path limitleads to a fractional anomalous diffusion equation. Our aim is to developnumerical schemes for the original kinetic model which works for the differentregimes, without being restricted by stability conditions of standard explicittime integrators. First, we propose some numerical schemes for the diffusionasymptotics; then, their extension to the anomalous diffusion limit is studied.In this case, it is crucial to capture the effect of the large velocities ofthe heavy-tailed equilibrium, so that some important transformations of theschemes derived for the diffusion asymptotics are needed. As a result, weobtain numerical schemes which enjoy the Asymptotic Preserving property in theanomalous diffusion limit, that is: they do not suffer from the restriction onthe time step and they degenerate towards the fractional diffusion limit whenthe mean free path goes to zero. We also numerically investigate the uniformaccuracy and construct a class of numerical schemes satisfying this property.Finally, the efficiency of the different numerical schemes is shown throughnumerical experiments.

Numerical schemes for kinetic equations in the diffusion and anomalous diffusion limits. Part I: the case of heavy-tailed equilibrium