An asymptotic-preserving scheme for the kinetic equation with anisotropic scattering: Heavy tail equilibrium and degenerate collision frequency

We design an asymptotic preserving (AP) scheme for the linear kinetic equation with anisotropic scattering that leads to a fractional diffusion limit. This limit may be attributed to two reasons: A heavy tail equilibrium or a degenerate collision frequency, both of which are considered in this paper. Our scheme builds on the ideas developed in [L. Wang and B. Yan, J. Comput. Phys., 312 (2016), pp. 157-174] but with two major variations. One is a new splitting of the system that accounts for the anisotropy in the scattering cross section by introducing two extra terms. We then showed, via detailed calculation, that the scheme enjoys a relaxed AP property as opposed to the one step AP for the isotropic scattering. Another contribution is for the degenerate collision frequency case, which brings in additional stiffness. We propose to integrate a "body" term, which appears to be the main component in the diffusion limit. This term is precomputed once with a prescribed accuracy, via a change of variable that alleviates the stiffness. Numerical examples are presented to validate its efficiency in both kinetic and fractional diffusion regimes.