Numerical solution of Boltzmann's equation

Abstract

This chapter presents the numerical solution of Boltzmann's equation. Boltzmann's equation describes the evolution of the one-particle distribution function f = f ( x, u, t ), where the vector x , with components x 1 , x 2 , x 3 , is the position vector; u , with components u 1 , u 2 , u 3 , is the velocity vector; and t is the time. A numerical treatment of Boltzmann's equation relies on a Monte-Carlo technique; some of these treatments are ingenuous and interesting, but none can be considered accurate. The distribution function f is approximated at each time step by two distinct discrete representations, once as a set of functional values at appropriate points and once as a set of coefficients in a Hermite expansion. The two representations are related by weighted Gaussian quadrature formulas. The main function of the Hermite representation is to provide a stable and accurate interpolation procedure for use in the evaluation of the collision term. The use of Hermite polynomials is suggested by the known properties of f , in a manner analogous to the choice of weights in Gaussian quadrature. The two representations of f are of equal significance.