Numerical solution of the Boltzmann Transport Equation

Abstract

A numerical model for solution of the linear Boltzmann Transport Equation is formulated. By applying the same techniques used in the derivation of the analytic equation, a discrete analog of the Boltzmann equation is derived for a finite cell in phase space. Initially undetermined coefficients in the analog are determined by requiring the numerical formulation to include properties (e.g., particle conservation) of the analytic equation. Terms occurring in the finite-cell analog are defined, and two treatments of the angular dependence are illustrated. A discrete ordinates representation is derived based on a connected straight-line-angular representation. This formulation maintains optical reciprocity and may be generalized. The second portion of the paper describes the systematic derivation of difference relations necessary to complete solution of the numerical formulation. Both representation schemes, based on assumed forms of particle fluxes in the cell, and characteristic schemes are examined. Difficulties encountered in extrapolations made by the use of representation schemes are clarified, and insight is gained for methods that can be used to prevent flux oscillations in numerical calculations.