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Parallel-in-time methods for partial differential equations (PDEs) have beenthe subject of intense development over recent decades, particularly fordiffusion-dominated problems. It has been widely reported in the literature,however, that many of these methods perform quite poorly foradvection-dominated problems. Here we analyze the particular iterativeparallel-in-time algorithm of multigrid reduction-in-time (MGRIT) fordiscretizations of constant-wave-speed linear advection problems. We focus oncommon method-of-lines discretizations that employ upwind finite differences inspace and Runge-Kutta methods in time. Using a convergence framework wedeveloped in previous work, we prove for a subclass of these discretizationsthat, if using the standard approach of rediscretizing the fine-grid problem onthe coarse grid, robust MGRIT convergence with respect to CFL number andcoarsening factor is not possible. This poor convergence and non-robustness iscaused, at least in part, by an inadequate coarse-grid correction for smoothFourier modes known as characteristic components.We propose an alternativecoarse-grid that provides a better correction of these modes. This coarse-gridoperator is related to previous work and uses a semi-Lagrangian discretizationcombined with an implicitly treated truncation error correction. Theory andnumerical experiments show the coarse-grid operator yields fast MGRITconvergence for many of the method-of-lines discretizations considered,including for both implicit and explicit discretizations of high order.Parallel results demonstrate substantial speed-up over sequentialtime-stepping.

Efficient multigrid reduction-in-time for method-of-lines discretizations of linear advection