An asymptotic preserving angular finite element based unified gas kinetic scheme for gray radiative transfer equations

Abstract

In order to mitigate the ray effects of the usual discrete ordinate (SN) method for the gray radiative transfer equations, a new numerical approach is constructed in this paper with the angular discretization by a linear finite element (FE) method and the spatial discretization by the method of unified gas kinetic scheme (UGKS). Different from the usual SN-based UGKS for which the propagation directions of photons are discretized with finite points, the angular-FE-based UGKS considers the linear combination of all propagation directions (i.e., the unit sphere) and induces a coupling between the discrete directions. Hence, the angular-FE-based UGKS can much mitigate the ray effects to some extent as was expected, as also shown numerically by a point source problem in this paper. At the same time, it is shown that the current scheme possesses the asymptotic preserving (AP) property, that is, in the optically thick regimes the current scheme can exactly capture the solution of the diffusion limit equation without requiring the cell size being smaller than the photon’s mean free path, while the solution in optically thin regimes can also be well resolved in a natural way. Various numerical experiments are included to validate the robustness, accuracy and AP property of the current scheme.