Finite difference methods for the Fokker-Planck equation

Abstract

Two methods are developed for numerical solution of the time-dependent Fokker-Planck equation. First, a finite difference scheme is constructed which has the property of simultaneously conserving total system mass, energy, and momentum throughout the interior of velocity space, thereby reducing nonconservation errors to boundary contributions only. Second, the Fokker-Planck collision operator is expanded in Legendre polynomials and reduced to an equivalent system of one-dimensional problems. The expansion is carried out to all orders and is mathematically exact; no terms are neglected or linearized. The resulting system of one-dimensional equations is given explicitly. Both these formulations are given for the case of two-dimensional azimuthally symmetric distribution functions. As a test of the methods, the thermal equipartition of an isotropic deuterium-electron plasma is computed numerically. Over a range of initial conditions from T e / T D = 0.01 to T e / T D = 100 the electron and ion temperatures obtained from the Fokker-Planck equation agree to within 6 % with those obtained from the Maxwellian model given by Spitzer.