Numerical approximation of kinetic Fokker-Planck equations with specular reflection boundary conditions

This work is devoted to the analysis of a numerical approximation to a general multi -dimensional kinetic Fokker-Planck (FP) equation with reaction and source terms and subject to specular reflection boundary conditions. This numerical approximation is based on splitting the kinetic FP model into a transport equation in space and a FP diffusive model in the velocity coordinates. The former is discretized by a Kurganov-Tadmor finite -volume scheme, while the latter is approximated by a generalized Chang & Cooper finite -volume method. Time integration is performed by a strong stability -preserving Runge-Kutta method where the reaction and source terms are accommodated with a Strang splitting technique and the use of a Magnus integrator. It is proved that the resulting numerical solution method is conservative and positive preserving, in the case where the continuous model has these properties, and it is second -order accurate in time and in phase space in the L1 -norm, subject to a CFL condition. Results of numerical experiments are reported that validate these theoretical results.