  We construct numerical schemes to solve kinetic equations with anomalous
diffusion scaling. When the equilibrium is heavy-tailed or when the collision
frequency degenerates for small velocities, an appropriate scaling should be
made and the limit model is the so-called anomalous or fractional diffusion
model. Our first scheme is based on a suitable micro-macro decomposition of the
distribution function whereas our second scheme relies on a Duhamel formulation
of the kinetic equation. Both are \emph{Asymptotic Preserving} (AP): they are
consistent with the kinetic equation for all fixed value of the scaling
parameter $\varepsilon >0$ and degenerate into a consistent scheme solving the
asymptotic model when $\varepsilon$ tends to $0$. The second scheme enjoys the
stronger property of being uniformly accurate (UA) with respect to
$\varepsilon$. The usual AP schemes known for the classical diffusion limit
cannot be directly applied to the context of anomalous diffusion scaling, since
they are not able to capture the important effects of large and small
velocities. We present numerical tests to highlight the efficiency of our
schemes.
