a parareal algorithm based on waveform relaxation

we report a new parallel iterative algorithm for time-dependent differential equations by combining the known waveform relaxation (wr) technique with the classical parareal algorithm. the parallelism can be simultaneously exploited in both sub-systems by wr and time by parareal. we also provide a sharp estimation on errors for the new algorithm. the iterations of parareal and wr are balanced to optimize the performance of the algorithm. furthermore, the parallel speedup and efficiency of the new approach are analyzed by comparing with the classical parareal algorithm and the wr technique, respectively. numerical experiments are carried out to verify the effectiveness of the theoretic work.