Parareal-based waveform relaxation method for semi-linear parabolic equation

We propose an efficient parareal-based waveform relaxation (WR) method in the two-stage iteration manner for the general semi-linear parabolic equation. The incorporation of the parareal iteration into the WR method allows for the parallel-in-time integration of the decoupled linear parabolic equations for each outer WR iteration. This novel approach may be regarded as a time-parallel accelerated WR method or as a variant of the parareal algorithm with varying propagators. In order to establish the convergence of this new method, we first revisit the convergence of the continuous case for the WR method of the semi-linear parabolic equation and present a new, more general error bound for the discrete case. Thereafter, we perform a convergence analysis of our parareal-based WR method for both continuous and discrete cases by introducing the perturbed systems. Finally, we provide numerical examples to validate the theoretical results and illustrate the efficiency of this method.