Understanding the effects of grid coarsening in space within the Parareal algorithm

Abstract

Among all Parallel-in-Time solutions that are available today, Parareal has the great advantage of not requiring heavy implementation work to be used within any already existent time-stepping solver. In the particular case of an explicit solver, space coarsening can be used to define a coarse solver, that provides interesting properties in terms of accuracy and time-parallel speedup. However, a major drawback of Parareal is that numerical instabilities can be expected for advection dominated problems, if some simulation parameters are not chosen wisely, hence a loss of parallel speedup. In this talk, we study the use of Parareal with space coarsening to solve the linear advection equation, by the mean of a Fourier analysis. The latter combines concepts from both Von Neuman analysis for space-time discretizations and Local Fourier Analysis for multi-grid methods. We identify the main effects of the grid coarsening on the convergence of Parareal, in order to bring some answers on the following questions: • Can Parareal with space-coarsening be stable when used on the advection problem ? • What is the influence of the chosen interpolation methods, and its order ? • Does the initial solution play a role in the convergence of the parallel-in-time algorithm ? Finally, a comparison to a bigger and more complex problem (Navier-Stokes equations) will be done to assess the generality of this analysis.