Multigrid reduction-in-time convergence for advection problems: A Fourier analysis perspective

TL;DR

This paper develops a Fourier analysis-based convergence theory for multigrid reduction-in-time (MGRIT) methods applied to advection problems, revealing why standard approaches often fail and suggesting ways to improve robustness.

Contribution

It introduces a local Fourier analysis framework for MGRIT applied to advection PDEs, providing insights into convergence issues and guiding the development of more robust algorithms.

Findings

Poor convergence linked to inadequate coarse-grid correction of characteristic Fourier modes.

Rediscretized coarse-grid operators in MGRIT are not robust against CFL number or coarsening factor.

Applying multigrid techniques from spatial problems can enhance MGRIT convergence.

Abstract

A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two-level variant of the iterative parallel-in-time method of multigrid reduction-in-time (MGRIT). This closed-form theory allows for new insights into the poor convergence of MGRIT for advection-dominated PDEs when using the standard approach of rediscretizing the fine-grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse-grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady-state advection-dominated PDEs. We apply this convergence theory to show that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re-purposed in the MGRIT context to develop more robust parallel-in-time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach.