Combined Hermite spectral-finite difference method for the Fokker-Planck equation

Abstract

In this paper, combined Hermite spectral-finite methods are proposed for numerical solution of the Fokker-Planck equation. The related physical problems are posed in a phase plane (x, v) with bounded position x and unbounded velocity v. The Hermite functions are employed to expand the velocity part of the solution, since (i) they form a complete system; (ii) they have correct natural boundary conditions in velocity space and (iii) they lead to the tridiagonal structure of the coupling system. The resulting system from the Hermite expansion is of hyperbolic type with source terms. An orthogonal transformation enables us to use the upwinding approach for the hyperbolic system. The convergence theory for the proposed schemes, both implicit and explicit, is established. For the explicit scheme, a generalized CFL condition is obtained. Numerical experiments are carried out to show the efficiency and accuracy of the numerical schemes.