Parallel Approximate Ideal Restriction Multigrid for Solving the S_N Transport Equations

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 + M, for d dimensions, P processors, and M angles. Moreover, an efficient scalable parallel sweep algorithm for use on general unstructured meshes remains elusive. Recently, an 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 the AMG methods (logarithmic in P) as well as the simplicity with which the 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. The 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.

Keywords:

• Transport
• multigrid
• sweep

Acknowledgments

This material is based upon work supported by the Department of Energy (DOE), National Nuclear Security Administration (NNSA), under award number DE-NA0002376. Established by Congress in 2000, NNSA is a semi-autonomous agency within the DOE responsible for enhancing national security through the military application of nuclear science. NNSA maintains and enhances the safety, security, reliability, and performance of the U.S. nuclear weapons stockpile without nuclear testing; works to reduce global danger from weapons of mass destruction; provides the U.S. Navy with safe and effective nuclear propulsion; and responds to nuclear and radiological emergencies in the U.S. and abroad.

The work of Ruipeng Li is performed under the auspices of the DOE by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344 (LLNL-JRNL-807107).

Notes

^a https://github.com/hypre-space/hypre

^b This could also be normalized, but we prefer to explicitly include some leading constant for each.