Parallel iteration across the steps of high-order Runge-Kutta methods for nonstiff initial value problems

Abstract

UvA-DARE (Digital Academic Repository) Parallel Iteration across the steps of high-order Runge-Kutta Methods for nonstiff initial value problems van der Houwen, P.J.; Sommeijer, B.P.; van der Veen, W.A.

Convergence aspects of step-parallel iteration of Runge-Kutta methods

“…In this paper, we shall allow j * to be greater than 1. As already observed in [11], the convergence analysis of (2.2) cannot be restricted to a local analysis of the iteration errors at a fixed point tn, but should be a global analysis where iteration errors at all preceding step points are involved. We shall distinguish two situations: (i) the predictor is based on iterates generated by the iteration scheme (3.1), and (ii) the iterates y(1) are generated independently, that is, the predictor is completely independent of the iteration scheme.…”

“…In a theoretical analysis, however, it seems not feasible to allow the parameters m and j * to be arbitrary functions of n, so that in deriving convergence results, m and j * are assumed to be constant. In our first investigations of step-parallel iterations schemes in [11,12], we hoped that sufficient robustness could already be obtained for j * = 1. We therefore analysed convergence only for j * = 1.…”

“…In [11,12] we discussed convergence of (3.1) for the case j * = 1. In this paper, we shall allow j * to be greater than 1.…”

“…There are several options for choosing the matrix B. For example, the case B = O (fixed point iteration) was studied in [11] and the resulting method was called the PIRKAS method (Parallel Iterated RK Across the Steps). In addition to parallelism across the steps, PIRKAS methods also have parallelism across the components of the iterates, because all components of F(¥~ j-~)) can also be evaluated in parallel.…”

“…This iteration method was called the PDIRK iteration method (Parallel Diagonalimplicit Iterated RK method). Following the ideas of Bellen and co-workers, a further level of parallelism was introduced in [11][12][13] by applying the PDIRK iteration scheme concurrently at a number of step points on the t-axis. In this paper, we shall analyse the convergence of these step-parallel PDIRK methods.…”