Journal of Computational Physics 161, 558–575 (2000) doi:10.1006/jcph.2000.6513, available online at http://www.idealibrary.com on The Spectral Collocation Method for the Kinetic Equation with the Nonlinear Two-Dimensional Coulomb Collisional Operator Ildar. K. Khabibrakhmanov ∗ and George. V. Khazanov† ∗ Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada, S7N 5E2; †Geophysical Institute and Department of Physics, University of Alaska, Fairbanks, Alaska, 99701 E-mail: ildar@isas62.usask.ca,khazanov@gi.alaska.edu Received April 2, 1999; revised March 20, 2000 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. c  2000 Academic Press Key Words: Fokker–Planck equation; spectral collocation; Coulomb collisions. CONTENTS 1. Introduction. 2. The Fokker–Planck operator. 3. Numerical results. 4. Conclusions. 1. INTRODUCTION Detailed knowledge of the charged particle distribution function is very important in many areas of plasma physics. It is quite common in applications to assume that the main 558 0021-9991/00 $35.00 Copyright c  2000 by Academic Press All rights of reproduction in any form reserved.