A Convergence Estimate of the Parareal Optimized Schwarz Waveform Relaxation Algorithm

We present and analyze in this paper a new space-time parallel method for solving evolution equations, i.e., the Parareal optimized Schwarz waveform relaxation (POSWR) algorithm. Since the classical Dirichlet transmission conditions inhibit the information exchange between subdomains to slow down the convergence speed of the Parareal Schwarz waveform relaxation (PSWR) algorithm, we introduce a class of optimized transmission conditions of Robin type to improve the convergence performance. We provide a convergence factor estimate based on the Laplace transform when the POSWR algorithm both with and without overlap is applied to the representative one-dimensional heat equation. Furthermore, we also analyze the optimized choice for the free parameter in the Robin transmission conditions to optimize the convergence behavior of the POSWR algorithm, which is different for the overlapping and nonoverlapping cases. Finally, we illustrate our theoretical analysis with several numerical experiments. We show that the new optimized algorithm even if without overlap converges much faster than the classical one, and the POSWR algorithm has different convergence behaviors in the overlapping and nonoverlapping cases.