A phase-space discontinuous Galerkin approximation for the radiative transfer equation in slab geometry

TL;DR

This paper develops and analyzes a symmetric interior penalty discontinuous Galerkin method for the second-order radiative transfer equation in slab geometry, demonstrating its accuracy, stability, and efficiency with adaptive refinement and error estimation.

Contribution

It introduces a novel DG scheme for the radiative transfer equation in slab geometry, including adaptive mesh refinement and error estimation techniques.

Findings

The method achieves high accuracy and stability across polynomial degrees.

Adaptive quad-tree grids effectively refine solutions in discontinuous regions.

Hierarchical and local averaging error estimators guide efficient adaptivity.

Abstract

We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically, for different polynomial degrees. For discretization, we employ quad-tree grids, which allow for local refinement in phase-space, and we show exemplary that adaptive methods can efficiently approximate discontinuous solutions. We investigate the behavior of hierarchical error estimators and error estimators based on local averaging.