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 ẏ(t)+Qy(t)=f(t). Outer WR iterations are defined by ẏk+1(t)+Dyk+1(t)=N1yk(t)+f(t), where Q=D−N1, and each iteration yk+1(t) is computed using an inner iterative process, based on an other splitting D=M−N2. 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−N1−N2 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.