Symmetry of the nonlinear collision operator matrix and new prospects in the moment method for solving the Boltzmann equation

Abstract

Traditionally, the moment method has been used in kinetic theory to calculate transport coefficients. Its application to the solution of more complicated problems runs into enormous difficulties associated with calculating the matrix elements of the collision operator. The corresponding formulas for large values of the indices are either lacking or are very cumbersome. In this paper relations between matrix elements are derived from very general principles, and these can be employed as simple recurrence relations for calculating all the nonlinear and linear anisotropic matrix elements from assigned linear isotropic matrix elements. Efficient programs which implement this algorithm are developed. The possibility of calculating the distribution function out to 8–10 thermal velocities is demonstrated. The results obtained open up prospects for solving many topical problems in kinetic theory.