PSEUDOSPECTRA OF WAVE-FORM RELAXATION OPERATORS

The performance of the waveform relaxation method for solving systems of ODEs depends largely on the choices that are made for splitting, si ze of time window, and preconditioning. Although it is known that supe rlinear convergence is obtained on finite time windows, the convergenc e may be slow in the first few iterations. We propose the use of pseud ospectra 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 an d overlapping on the rate of convergence. We may also use this to esti mate a suitable size of the time window. Numerical experiments perform ed on a system of ODEs arising from the discretization of an advection -diffusion equation confirm the validity of the obtained estimates. (C ) 1998 Elsevier Science Ltd. All rights reserved.