Waveform Relaxation methods have been proposed and used to solve large systems of ordinary differential equations (ODEs) on parallel computers. They split the system of ODEs into subsystems and evolve subsystems independently over a short period of time before communicating results to neighboring subsystems. The convergence of Waveform Relaxation methods however is slow and often the parallel execution gain is not sufficient to compensate for the slow convergence. Thus Waveform Relaxation is only used in practice whenever there is no other alternative, for example because of memory restrictions or to solve equations with coupling terms for which no other numerical method is known today.

However there is a fundamental reason why Waveform Relaxation methods are slow and the performance of the method can be improved by orders of magnitudes with a small change to the algorithm. We show for a large system of ODEs coming from a discretization of a convection reaction diffusion equation why Waveform Relaxation is slow and how to speed up the convergence rate. While the classical method reduces the initial error of order 1 in five iterations to 0.25, the slightly modified algorithm reduces the error to 10e(-8) in the same number of iterations.

MINISIMPOSIUM SESSION: 18. Waveform Relaxation Techniques