Parareal Schwarz Waveform Relaxation Methods

Solving an evolution problem in parallel is naturally undertaken by trying to parallelize the algorithm in space, and then still follow a time stepping method from the initial time t = 0 to the final time t = T . This is especially easy to do when an explicit time stepping method is used, because in that case the time step for each component is only based on past, known data, and the time stepping can be performed in an embarrassingly parallel way. If one uses implicit time stepping however, one obtains a large system of coupled equations, and thus the linear or non-linear solver needs to be parallelized, e.g. using a domain decomposition method. Over the last decades, people have however also tried to parallelize algorithms in the time direction. One example is Womble’s algorithm [22], where the systems arising from an implicit time discretization are solved using an iterative method, and the iteration of the next time level is started, before the iteration on the current time level has converged. It is then possible to iterate several time levels simultaneously, but the possible gain using a parallel computer is only small, see for example [3]. A different approach to obtain small scale parallelism in time is to use predictor-corrector methods, where the prediction step and the correction step can be performed by two (or several) processors in parallel, if organized properly. An entire class of such methods has been proposed in [19], and good small scale parallelism can be achieved.