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.

Ordinary Differential Equations (acm G.1.7)
Department of Numerical Mathematics [NM]

van der Houwen, P.J, & Sommeijer, B.P. (1996). CWI contributions to the development of parallel Runge-Kutta methods. Department of Numerical Mathematics [NM]. CWI.