An iterative splitting method via waveform relaxation

Abstract

This paper explores a new numerical strategy for a closed formulation of iterative splitting methods and their embedding in classical waveform-relaxation methods. Since iterative splitting has been developed in several papers, an abstract framework that relates these methods to other classical splitting methods would be useful and is needed. Here, we present an embedding of the iterative splitting method in the waveform-relaxation and exponential splitting methods. While we can use the theoretical background of the classical schemes, a simpler iterative splitting analysis is obtained. This is achieved by basing the analysis on semigroup and fixpoint schemes. Our approach is illustrated with numerical results obtained on differential equations with constant and time-dependent coefficients.

Keywords:

• iterative splitting method
• waveform-relaxation method
• pade approximations

2000 AMS Subject Classifications :

• 65M15
• 65L05
• 65M71

Notes

Please note that the dependencies of are suppressed for the sake of simplicity.

As we will see, there is an exception to this.

In fact, the order of the approximation is not of much importance if we fulfil a sufficient number of iterations. In the case of u ^−1/I =0, we have the exception that a step in the A-direction is done while B is left out. The error of this step certainly vanishes after a few iterations, but mostly after only one iteration.

A point in favour of the iterative splitting scheme is that it also takes into account the fact that AB splitting may be used alongside the high-order methods alluded to but cannot maintain the order if [A, B]≠0, while the iterative splitting scheme re-establishes the maximum order of the scheme after a sufficient number of iterations have been performed.

The code for both methods is kept in the simplest possible form.