A practical difference scheme for Fokker-Planck equations☆

A practical finite difference scheme for initial value problems of Fokker-Planck equations has been studied. In addition to satisfying the conditions of convergence and unconditional stability, this scheme provides numerical solutions which preserve some of the more important intrinsic properties of the original partial differential equation. In particular, the solutions are non-negative, particle conserving in the absence of external sources or sinks, and exact representations of the analytic solution upon equilibration. Furthermore, coupled with variable mesh size, this scheme actually significantly reduces the number of mesh points required with no loss of accuracy.