Pseudospectra of waveform relaxation operators

The performance of the waveform relaxation method for solving systems of ODEs depends largely on the choices that are made for splitting, size of time window, and preconditioning. Although it is known that superlinear convergence is obtained on finite time windows, the convergence may be slow in the first few iterations. We propose the use of pseudospectra to analyze the convergence ratio of the first few iterations when waveform relaxation is applied to linear systems of ODEs. Through pseudospectral radii, we can examine the effect of preconditioning and overlapping on the rate of convergence. We may also use this to estimate a suitable size of the time window. Numerical experiments performed on a system of ODEs arising from the discretization of an advection-diffusion equation confirm the validity of the obtained estimates.