  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.
