Parallelism Across the Steps in Iterated Runge-Kutta Methods for Stiff InitialValue Problems

摘要

For the parallel integration of stiff initial value problems (IVPs), three mainapproaches can be distinguished: approaches based on 'parallelism across the problem', 'parallelism across the method' and on 'parallelism across the steps'. The first type of parallelism does not require special integration methods and can be exploited within any available IVP solver. The method-parallel approach received some attention in the case of Runge-Kutta based methods. For these methods, the required number of processors is roughly half the order of the generating Runge-Kutta method and the speed-up with respect to a good sequential IVP solver is about a factor 2. The third type of parallelism (step-parallelism) can be achieved in any IVP solver based on predictor-corrector iteration. Most step-parallel methods proposed so far employ a large number of processors, but lack the property of robustness, due to a poor convergence behavior in the iteration process. Hence, the effective speed-up is rather poor. The step-parallel iteration process proposed in the present paper is less massively parallel, but turns out to be sufficiently robust to achieve speed-up factors up to 10 with respect to the best sequential codes.