A Unified Framework Based on Operational Calculus for the Convergence Analysis of Waveform Relaxation Methods

Waveform relaxation methods are iterative methods for the solution of systems of ordinary differential equations. The convergence analysis of waveform relaxation methods traditionally uses the theory of Volterra convolution equations. More specifically, the convergence theory is typically based on a theorem of Paley and Wiener that gives a condition for the solution of a linear Volterra convolution equation to be bounded. Extensions of this theorem to discrete convolution equations and vector-valued problems have been described in the literature. In this note, it is shown that the same results can be derived by an alternative approach based on operational calculus. An operational calculus defines what is meant by a function of an operator. A spectral mapping theorem then relates the spectrum of the resulting operator to the spectrum of the original operator. In this paper, the Dunford-Taylor operational calculus for scalar analytic functions is extended to matrix-valued analytic functions. Using the corresponding spectral mapping theorem, it is then straightforward to analyze the convergence of a large number of waveform relaxation algorithms. The theory is applied to the analysis of continuous waveform relaxation and to the analysis of discrete waveform relaxation based on general linear methods, both for initial value and timeperiodic systems of ordinary differential equations.