New Quarter-Sweep-Based Accelerated Over-Relaxation Iterative Algorithms and their Parallel Implementations in Solving the 2D Poisson Equation

This thesis deals with iterative methods for solving the Poisson equation, which is a representative of partial diferential equations. The research considers different techniques and strategies in over-relaxation theory. The over-relaxation methods are easy to implement on a computer and exible in management of the rate of convergence. Recent research in this area is related to different variations and applications of Successive Over-Relaxation (SOR) and Accelerated Over-Relaxation (AOR) methods. Three types of fnite-diference schemes are in the base of the full-sweep (FS),half-sweep (HS), and quarter-sweep (QS) approaches, considered in this research. Among them, the QS approach is shown to be the fastest and the most eco-nomical, achieving satisfactory result with less number of operations. Another approach to speed up the convergence is grouping of iteration points into a single iteration unit. Implemented with the fnite-diference schemes mentioned above,this approach produces Explicit Group (EG), Explicit Decoupled Group (EDG),and Modifed Explicit Group (MEG) methods. While all the above mentioned methods were implemented with SOR, among them, the QS point and MEG methods have never been implemented with AOR before. The main objective of the thesis is to develop new sequential and parallel iterative methods that will be faster and more e�cient as compared to the existing meth-ods. Eventually, new AOR QS and AOR MEG iterative methods are proposed. The experimental results and numerical complexity analysis have shown the new methods to be much faster than the existing counterparts. With respect to the AOR EDG method, which is the fastest counterpart, the total improvement in terms of execution time is about 74%.Parallel implementations of these methods are very important, since high perfor-mance computing has become main supportive technology of scientific research. Newly developed parallel AOR QS and AOR MEG methods for distributed mem-ory parallel machine are shown to be eficient for large sparse matrices, which occur in large sizes of problem. The parallel strategies used in the new algorithms are based on the message latency minimization and processor-independent iterations.

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