Analysis of a New Accelerated Waveform Relaxation Method Based on the Time-Parallel Algorithm

In this paper, we propose a new accelerated waveform relaxation (WR) method based on a time-parallel algorithm to solve the general system of ordinary differential equations (ODEs). It is well known that the WR method decouples or linearizes large-scale complex systems into simple subsystems, which in most cases can be computed in parallel in each iteration. To accelerate the calculation of the WR iteration, we apply a time-parallel approach: the Parareal algorithm, to solve the subsystems in each iteration. It can be thought of as a kind of space-time parallel method. According to different WR types, we present convergence analysis of the accelerated WR methods for the time-continuous case and for the time-discrete case with different discrete schemes. Besides, the speedup analysis of the proposed algorithms is also provided. Finally, numerical experiments are carried out to verify the effectiveness of the theoretical works.