A Robust Multigrid Preconditioner for $S_N$DG Approximation of Monochromatic, Isotropic Radiation Transport Problems

Abstract

We introduce a new stabilization for the $S_N$DG (discrete ordinate discontinuous Galerkin) approximation of monochromatic radiation transport, and argue that solutions converge to solutions to the LDG method of Cockburn and Shu in the thick diffusion limit. Then, we develop a multilevel scheme for this discretization. Nonoverlapping Schwarz smoothers are based on solving local radiation transport problems for each grid cell. The ideal version of the smoother uses sweeps forward and backward in the direction of the diagonal of each octant. A simplified version of the multilevel solver runs on each cell in parallel, but lacks robustness in vacuum. In extensive tests we verify that the number of iterations for a given gain in accuracy is independent of the mesh size and the scattering cross section.