Space–time streamline upwind Petrov–Galerkin methods for the Boltzmann transport equation

Abstract

This paper presents a space–time streamline upwind Petrov–Galerkin formulation for the time-dependent Boltzmann transport equation. Conventional linear space–time formulations typically take the distance across space–time elements as the representative distance across which dissipation is introduced. The proposed method however, calculates the representative distances based on the gradient of the solution in space–time. Therefore, the method in its most general form is non-linear. The calculation of distances based on the solution gradient means that even if small time-steps are used the method will attempt to optimise the numerical accuracy while eliminating any spurious oscillations.Unlike conventional non-linear Petrov–Galerkin formulations the proposed class of methods introduces dissipation only in the streamline direction (in space–time). Conventional non-linear discontinuity capturing schemes add dissipation in the steepest gradient direction. In order to demonstrate the accuracy and robustness (in terms of suppressing spurious oscillations), the method is applied to a series of one-group, 2-D, fixed source, steady-state and time-dependent radiation transport problems.