Using the classical waveform relaxation method for solving large systems of ordinary differ- ential equations in parallel leads to a slow and non-uniform convergence over the time interval for which the equations are integrated. To improve performance, a new class of waveform relaxation methods known as optimized waveform relaxation algorithms were developed which greatly improve the convergence by using new transmission conditions. These conditions are responsible for the exchange of information between the subsystems which we obtain by decomposing the original large system of differential equations. The waveform relaxation methods were first introduced for solving VLSI circuit simulations, and have been extended to time dependent PDE’s. This talk consists of two parts as follows.

Part-II: in this second part we present some theories for the optimized waveform relaxation methods, including the solution of a class of min-max optimization problems, which plays a central role of finding the crucial parameters for the transmission conditions.