The Vlasov-Poisson-Fokker-Planck System with Uncertainty and a One-dimensional Asymptotic Preserving Method

We develop a stochastic Asymptotic Preserving (s-AP) scheme for the Vlasov-PoissonFokker-Planck (VPFP) system in the high field regime with uncertainty based on the generalized Polynomial Chaos Stochastic Galerkin framework (gPC-SG). We first prove that, for a given electric field with uncertainty, the regularity of initial data in the random space is preserved by the analytical solution at later time, which allows us to establish the spectral convergence of the gPC-SG method. We follow the framework developed in [15] to numerically solve the resulting system in one space dimension, and show formally that the fully discretized scheme is s-AP in the high field regime. Numerical examples are given to validate the accuracy and s-AP properties of the proposed method.