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We consider the parallel time integration of the linear advection equationwith the Parareal and two-level multigrid-reduction-in-time (MGRIT) algorithms.Our aim is to develop a better understanding of the convergence behaviour ofthese algorithms for this problem, which is known to be poor relative to thediffusion equation, its model parabolic counterpart. Using Fourier analysis, wederive new convergence estimates for these algorithms which, in conjunctionwith existing convergence theory, provide insight into the origins of this poorperformance. We then use this theory to explore improved coarse-gridtime-stepping operators. For several high-order discretizations of theadvection equation, we demonstrate that there exist non-standard coarse-gridtime stepping operators that yield significant improvements over the standardchoice of rediscretization.

On selecting coarse-grid operators for Parareal and MGRIT applied to linear advection