  In this work, we propose some numerical schemes for linear kinetic equations
in the diffusion and anomalous diffusion limit. When the equilibrium
distribution function is a Maxwellian distribution, it is well known that for
an appropriate time scale, the small mean free path limit gives rise to a
diffusion type equation. However, when a heavy-tailed distribution is
considered, another time scale is required and the small mean free path limit
leads to a fractional anomalous diffusion equation. Our aim is to develop
numerical schemes for the original kinetic model which works for the different
regimes, without being restricted by stability conditions of standard explicit
time integrators. First, we propose some numerical schemes for the diffusion
asymptotics; then, their extension to the anomalous diffusion limit is studied.
In this case, it is crucial to capture the effect of the large velocities of
the heavy-tailed equilibrium, so that some important transformations of the
schemes derived for the diffusion asymptotics are needed. As a result, we
obtain numerical schemes which enjoy the Asymptotic Preserving property in the
anomalous diffusion limit, that is: they do not suffer from the restriction on
the time step and they degenerate towards the fractional diffusion limit when
the mean free path goes to zero. We also numerically investigate the uniform
accuracy and construct a class of numerical schemes satisfying this property.
Finally, the efficiency of the different numerical schemes is shown through
numerical experiments.
