An Angular Multigrid Preconditioner for the Radiation Transport Equation with Forward-Peaked Scatter

TL;DR

This paper extends an angular multigrid preconditioner for the Boltzmann transport equation to handle general Legendre expansions of forward scatter, improving efficiency and reducing computational effort in anisotropic scattering problems.

Contribution

The paper introduces an extension of the multigrid preconditioner to non-Fokker-Planck models with Legendre expansions, enhancing its applicability to complex scattering scenarios.

Findings

Fewer iterations needed compared to single-mesh methods

Reduced computational effort demonstrated in test problems

Effective for highly anisotropic scatter with lowered scatter order

Abstract

In a previous paper (Lathouwers and Perk\'o, 2019) we have developed an efficient angular multigrid preconditioner for the Boltzmann transport equation with forward-peaked scatter modeled by the Fokker-Planck approximation. The discretization was based on a completely discontinuous Galerkin finite element scheme both for space and angle. The scheme was found to be highly effective on isotropically and anisotropically refined angular meshes. The purpose of this paper is to extend the method to non-Fokker-Planck models describing the forward scatter by general Legendre expansions. As smoother the standard source iteration is used whereas solution on the coarsest angular mesh is effected by a special sweep procedure that is able to solve this problem with highly anisotropic scatter using only a small number of iterations. An efficient scheme is obtained by lowering the scatter order in the multigrid preconditioner. A set of test problems is presented to illustrate the effectivity of the method, i.e. in less iterations than the single-mesh case and more importantly with reduced computational effort.