An analysis of convergence for two-stage waveform relaxation methods

This paper consider a two-stage (or inner/outer) strategy for waveform relaxation (WR) iterations, applied to initial value problems for linear systems of ordinary differential equations (ODEs) in the form y(t) + Qy(t) = f (t). Outer WR iterations are defined by y (k+1)(t) + Dyk+1(t) = N1 y(k) (t) + f (t), where Q = D - N-1, and each iteration y(k+1) (t) is computed using an inner iterative process, based on an other splitting D = M - N-2. Each ODE is then discretized by means of Theta method. For an M-matrix Q we prove that the method converges under the assumption that the whole splitting Q = M - N-1 - N-2 is an M-splitting, independently of the number of inner iterations. Moreover, some comparison results are given in order to relate the ratio of convergence of the whole inner/outer process both to the number of inner iterations actually done and to discretization parameters h and theta. Finally numerical experiments are presented. (C) 2004 Elsevier B.V. All rights reserved.