Coupling Parareal with Optimized Schwarz waveform relaxation for parabolic problems

Duc Quang Bui (1) , Caroline Japhet (1, 2) , Yvon Maday (3, 4) , Pascal Omnes (5, 1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
3 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
4 IUF - Institut universitaire de France
5 CEA - Commissariat à l'énergie atomique et aux énergies alternatives

We propose and analyse a parallel method, both in the time and space directions, that couples the Parareal algorithm with the Optimized Schwarz waveform relaxation (OSWR) method, with only few OSWR iterations in the fine propagator and with a simple coarse propagator deduced from the Backward Euler method. The analysis of this coupled method is presented for a one-dimensional advection-reaction-diffusion equation. We prove a general convergence result for this method via energy estimates. Numerical results for two-dimensional advection-diffusion problems and for a diffusion equation with strong heterogeneities are presented, to illustrate the performance of the coupled Parareal-OSWR algorithm.

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Duc Quang Bui, Caroline Japhet, Yvon Maday, Pascal Omnes. Coupling Parareal with Optimized Schwarz waveform relaxation for parabolic problems. 2021. ⟨hal-03167571v1⟩