Waveform relaxation as a dynamical system

Abstract

In this paper the properties of waveform relaxation are studied when applied to the dynamical system generated by an autonomous ordinary differential equation. In particular, the effect of the waveform relaxation on the invariant sets of the flow is analysed. Windowed waveform relaxation is studied, whereby the iterative technique is applied on successive time intervals of length T T and a fixed, finite, number of iterations taken on each window. This process does not generate a dynamical system on R + \mathbb {R}^+ since two different applications of the waveform algorithm over different time intervals do not, in general, commute. In order to generate a dynamical system it is necessary to consider the time T T map generated by the relaxation process. This is done, and C 1 C^1 -closeness of the resulting map to the time T T map of the underlying ordinary differential equation is established. Using this, various results from the theory of dynamical systems are applied, and the results discussed.