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Numerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part II: Degenerate Collision Frequency

In this work, which is the continuation of [SIAM J. Sci. Comput., 38 (2016), pp. A737--A764], we propose numerical schemes for linear kinetic equations 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 solving this 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.