Multigrid Reduction‐In‐Time Convergence for Advection Problems: A Fourier Analysis Perspective

Abstract

ABSTRACTA 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.