Research And Multi-step Iterative Technique Of Splitting Waveform Relaxation Method

The solution of differential equation is the core of modern large-scale scientific engineering calculation. With the rapid development of computer technology, the scale of solving the problem is larger and larger. However, as the effective method of the iterative method for solving large scale problems, it has also become one of the most important methods of solving large differential equation. Waveform relaxation method as a dynamic iterative method has attracted much attention. From the beginning of the classical iterative methods to fast relaxation iteration method, the alternative iteration method, two-stage iteration method and so on, they are all based on the studying of convergence of the iterative matrix by splitting the coefficient, then realize the solution of problem, and apply it to the corresponding actual condition.This article is based on solving the waveform relaxation method of differential equations. It has combined two-stage iterative method with splitting method, studied steady and unsteady two-stage waveform relaxation iteration method, given the convergence theory of two-stage waveform relaxation method when the coefficient matrix and corresponding division meet certain conditions, enriched the theoretical research of the iterative method. At the same time, this article studied the comparison theory according to two influencing factors, the step size hθ and the number of inner iterations P, particularly studied the comparison theory when the coefficient matrix is a special matrix and the corresponding division is effectively split.The main structure of this article as follows:The first part is introduction. It has mainly introduced the background, main ideas, the history of development and research situation of waveform relaxation method.The second part is preliminaries. It has introduced some definitions and lemmas which provide the evidences of the following proving.The third part is the main part of this article. It has studied the two-stage waveform relaxation method, analyzed the convergence theory of steady and unsteady iteration method. It has analyzed the convergence problem when coefficient matrix is special matrix and the matrix split is effective.The fourth part is also the main part. It has studied the comparison theory of two-stage waveform relaxation method, also both steady and unsteady. It has analyzed the comparison theory of the iteration method when the step size hθ and inner iterations are respectively changed and the coefficient matrix is a special matrix. and the splitting is effective.The fifth part is numerical examples. The numerical examples show the validity of the theoretical results in the third and fourth parts.The sixth part is the summary and prospect. The paper has summed up and put forward the greater space of study and application prospect.