Accuracy increase in waveform Gauss Seidel

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Abstract
The traditional approach for solving large dynamical systems is time consuming. Waveform relaxation, an iterative technique for solving systems of differential equations, can be used to reduce the processing time. It has been shown to converge superlinearly on finite intervals. In this paper, the order of accuracy of solutions generated by a relaxation approach, the waveform Gauss-Seidel method, is discussed. In this approach, a directed graph, called a dependency graph, is used to indicate the coupling relations among all components. The relation between the accuracy increase after each Gauss-Seidel iteration and the lengths of ascending chains (simple directed paths) in cycles in the dependency graph is discussed. It is proved that the cycle in the dependency graph which has the minimum ratio of its length to its number of ascending chains, determines the average accuracy increase. Effective use of the waveform Gauss-Seidel method depends on the ordering of the components. The result in this paper provides a basis for selecting the ordering. 2 refs.

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