A hybrid algorithm based on parareal and Schwarz waveform relaxation

In this paper, we present a hybrid algorithm based on parareal and Schwarz waveform relaxation (SWR) for solving time dependent partial differential equations. The parallelism can be simultaneously realized in the time direction by using a parareal and in the space direction via SWR. We give a convergence analysis for the hybrid algorithm for a 1D model problem, the reaction-diffusion equation. Weak scaling of the algorithm in terms of both the number of space subdomains and the number of paralleled time intervals were investigated via theoretical analysis and numerical experiments.