An analysis of convergence for two-stage waveform relaxation methods

Abstract

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)+Dy k+1(t)=N 1y 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 θ. Finally numerical experiments are presented.