AIR algebraic multigrid for a space-time hybridizable discontinuous Galerkin discretization of advection(-diffusion)

TL;DR

This paper explores the use of AIR algebraic multigrid as an effective preconditioner for space-time HDG discretizations of advection-dominated flows, demonstrating robustness and scalability in various settings.

Contribution

It introduces AIR as a preconditioner for space-time HDG discretizations of advection problems, showing its effectiveness and explaining its robustness through geometric coarsening analysis.

Findings

AIR preconditioner improves convergence for advection-diffusion problems

Effective on fixed and time-dependent domains with adaptive mesh refinement

Provides scalable solutions for hyperbolic and advective problems

Abstract

This paper investigates the efficiency, robustness, and scalability of approximate ideal restriction (AIR) algebraic multigrid as a preconditioner in the all-at-once solution of a space-time hybridizable discontinuous Galerkin (HDG) discretization of advection-dominated flows. The motivation for this study is that the time-dependent advection-diffusion equation can be seen as a "steady" advection-diffusion problem in $(d+1)$-dimensions and AIR has been shown to be a robust solver for steady advection-dominated problems. Numerical examples demonstrate the effectiveness of AIR as a preconditioner for advection-diffusion problems on fixed and time-dependent domains, using both slab-by-slab and all-at-once space-time discretizations, and in the context of uniform and space-time adaptive mesh refinement. A closer look at the geometric coarsening structure that arises in AIR also explains why AIR can provide robust, scalable space-time convergence on advective and hyperbolic problems, while most multilevel parallel-in-time schemes struggle with such problems.