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-Poisson-Fokker-Planck 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 a later time, which allows us to establish the spectral convergence of the gPC-SG method. We follow the framework developed in [S. Jin and L. Wang, Acta Math. Sci., 31 (2011), pp. 2219-2232] 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.