Nonsymmetric Algebraic Multigrid Based on Local Approximate Ideal Restriction (lAIR)

TL;DR

This paper introduces lAIR, a novel algebraic multigrid method tailored for nonsymmetric matrices, demonstrating robust and scalable convergence for complex advection-diffusion-reaction systems, especially on unstructured meshes and high-order discretizations.

Contribution

lAIR extends classical AMG to nonsymmetric matrices using local ideal restriction approximation, improving robustness and parallelizability for complex PDE discretizations.

Findings

Robust convergence on unstructured meshes and high-order finite elements.

Effective for both steady-state and time-dependent advection-diffusion-reaction problems.

Highly parallelizable algorithm avoiding communication-intensive cycles.

Abstract

Algebraic multigrid (AMG) solvers and preconditioners are some of the fastest numerical methods to solve linear systems, particularly in a parallel environment, scaling to hundreds of thousands of cores. Most AMG methods and theory assume a symmetric positive definite operator. This paper presents a new variation on classical AMG for nonsymmetric matrices (denoted lAIR), based on a local approximation to the ideal restriction operator, coupled with F-relaxation. A new block decomposition of the AMG error-propagation operator is used for a spectral analysis of convergence, and the efficacy of the algorithm is demonstrated on systems arising from the discrete form of the advection-diffusion-reaction equation. lAIR is shown to be a robust solver for various discretizations of the advection-diffusion-reaction equation, including time-dependent and steady-state, from purely advective to purely diffusive. Convergence is robust for discretizations on unstructured meshes and using higher-order finite elements, and is particularly effective on upwind discontinuous Galerkin discretizations. Although the implementation used here is not parallel, each part of the algorithm is highly parallelizable, avoiding common multigrid adjustments for strong advection such as line-relaxation and K- or W-cycles that can be effective in serial, but suffer from high communication costs in parallel, limiting their scalability.