Stable time-parallel integration of advection dominated problems using Parareal with space coarsening

A common idea in the PinT community is that Parareal, one of the most popular time-parallel algorithm, is often numerically unstable when applied to hyperbolic problems, such as the advection equation. This has been both numerically observed and studied theoretically in the case of an implicit time integrator as a coarse solver (see, e.g, [1]). Such results could discourage the application of Parareal to Computational Fluid Dynamics (CFD) problems, especially for high Reynolds numbers (see, e.g, [2]). On the contrary, a recent application of Parareal with spatial coarsening to an explicit CFD solver [3] has shown that not only a stable numerical integration was obtained, but also that the Reynolds number played a minor role in the convergence behaviour, compared to other parameters of the parallel-in-time algorithm. Hence, in this talk, we present numerical experiments related to the application of Parareal with spatial coarsening to the one-dimensional advection- diffusion problem. We investigate the influence of several parameters on the convergence (importance of the diffusion term, spatial resolution, order of interpolation, regularity of the initial solution, time-slice length, nonlinearity,...). We advocate that "a high Reynolds number" is not a good enough reason for not using Parareal, and that a stable and efficient parallel in time integration can be made possible, even for highly advective problems, provided that important algorithmic components are carefully chosen