Convergence of a difference scheme for the Vlasov-Poisson-Fokker-Planck system in one dimension

A finite difference scheme is proposed for the periodic one-dimensional Vlasov-Poisson-Fokker-Planck system which is related to the scheme in [C. Cheng and G. Knorr, J. Comp. Phys., 22 (1976), pp. 330-351]. The density f (t,x,v) and the electric field E (t,x) are shown to converge to the exact solution values in L-2 and L-infinity, respectively. The rate of convergence is essentially first order. This scheme has been implemented in [J. Schaeffer, Center for Nonlinear Analysis Research Report 97-NA-004, Carnegie Mellon University, 1997] and compares favorably with a random particle method.