An asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck equation

We propose a numerical method for a Vlasov-Poisson-Fokker-Planck model and prove quantitative results ensuring that it is Asymptotic-Preserving for the linearized model in both the macroscopic and the long time regime. We illustrate these results with various numerical experiments in which we observe, among others, transition phase between macroscopic and long time behavior as well as oscillations and instability phenomena.