CWI contributions to the development of parallel Runge-Kutta methods

This paper will concentrate on contributions of CWI to the development of parallel Runge-Kutta (RK) methods. We shall describe two approaches to construct such methods. In both approaches, a conventional implicit RK method is used as a corrector equation whose solution is approximated by an iterative method. In the first approach, the iteration method uses a fixed number of iterations without solving the corrector. Assuming that a one-step predictor is used, this approach again results in an RK method, however, an RK method possessing a lot of intrinsic parallelism. In the second approach, the corrector is solved by modified Newton iteration and the linear systems arising in each Newton iteration are solved by a parallel iteration process which is tuned to the special form of these linear systems. Furthermore, we apply the parallel iteration process in a step-parallel fashion which further enhances the amount of parallelism. Finally, the application of parallel RK methods within the framework of waveform relaxation is briefly discussed.