Parallel Approximate Ideal Restriction Multigrid for Solving the S$_N$ Transport Equations

TL;DR

This paper introduces a parallel algebraic multigrid method based on approximate ideal restriction (pAIR) as a scalable, robust alternative to traditional parallel sweep algorithms for solving the $S_N$ transport equations, especially on unstructured meshes.

Contribution

The paper demonstrates the effectiveness of pAIR as a scalable, black-box solver for $S_N$ transport equations, providing an alternative to traditional parallel sweep methods on structured and unstructured meshes.

Findings

pAIR is robust and scalable for $S_N$ transport equations.

pAIR performs similarly on structured and unstructured meshes.

Traditional sweeps are faster on structured meshes, but pAIR offers a versatile alternative.

Abstract

The computational kernel in solving the $S_N$ transport equations is the parallel sweep, which corresponds to directly inverting a block lower triangular linear system that arises in discretizations of the linear transport equation. Existing parallel sweep algorithms are fairly efficient on structured grids, but still have polynomial scaling, $P^{1/d}$ for $d$ dimensions and $P$ processors. Moreover, an efficient scalable parallel sweep algorithm for use on general unstructured meshes remains elusive. Recently, a classical algebraic multigrid (AMG) method based on approximate ideal restriction (AIR) was developed for nonsymmetric matrices and shown to be an effective solver for linear transport. Motivated by the superior scalability of AMG methods (logarithmic in $P$) as well as the simplicity with which AMG methods can be used in most situations, including on arbitrary unstructured meshes, this paper investigates the use of parallel AIR (pAIR) for solving the $S_N$ transport equations with source iteration in place of parallel sweeps. Results presented in this paper show that pAIR is a robust and scalable solver. Although sweeps are still shown to be much faster than pAIR on a structured mesh of a unit cube, pAIR is shown to perform similarly on both a structured and unstructured mesh, and offers a new, simple, black box alternative to parallel transport sweeps.