Asymptotic preserving and uniformly conditionally stable finite difference schemes for kinetic transport equations

In this work, uniformly unconditionally stable first and second order finite difference schemes are developed for kinetic transport equations in the diffusive scaling. We derived an approximate evolution equation for the macroscopic density, from the formal solution of the distribution function, which is then discretized by following characteristics for the transport part with a backward finite difference semi-Lagrangian approach, while the diffusive part is discretized implicitly. The resulting schemes can be shown to be asymptotic preserving (AP) in the diffusive limit. Uniformly unconditional stabilities are verified from a Fourier analysis. Numerical experiments, including high dimensional problems, have demonstrated the corresponding orders of accuracy both in space and in time, uniform stability, AP property, and good performances of our proposed approach.