Numerical approximation of the spatially homogeneous Fokker–Planck–Landau equation

摘要

A numerical method is developed for the spatially homogeneous Fokker–Planck–Landau (FPL) equation for the case of Coulomb collisions. The equation is put into a form so that finite difference methods for parabolic type PDE’s can be applied. Through a transformation of variables the velocity space computation is reduced to a bounded domain, and finite differencing is done with a well defined, exact, boundary condition corresponding to |v| at infinity. For the discretization in time implicit differencing is used. An analysis is carried out on the discrete approximation as applied to a linear FPL equation, and for the linear equation it is proved that the numerical method is unconditionally stable and convergent. Computational work is then done that demonstrates the unconditional stability and accuracy of the numerical method for the nonlinear FPL equation.