The Spectral Collocation Method for the Kinetic Equation with the Nonlinear Two-Dimensional Coulomb Collisional Operator

Abstract

The spectral collocation method is used for numerical solution of the Fokker–Planck equation with nonlinear integro-differential coulomb collisional operator. The spectral collocation method in general gives superior results to the usually employed finite difference method approximation. High order approximation of the integro-differential operator by the spectral collocation is able to provide highly accurate results on sparse grids. Approximation of the boundary conditions of the problem is very straightforward and natural. The method is also capable of easily accounting for the physically important conservation properties of the system. In this article the details of the numerical implementation of the Fokker–Planck equation solver with Coulomb collisional operator are discussed. Some test results are presented and certain limitations of the implementation are discussed. The method is applied to the problem of plasma heating by superthermal radiation. The self-similar solution is obtained for this case.