New Parallel Iterative Methods For The Solution Of Telegraph Equations
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Date
2015-08
Authors
LEE MING, KEW
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Abstract
Many applications in applied mathematics and engineering involve partial differential equations. Discretizing partial differential equations using some discretizing schemes based on finite difference or finite element methods often leads to a large and sparse system of linear equations. These linear equations are usually extremely difficult or impractical to analytically solve due to the complexity of the problem. Consequently, much effort has been devoted to obtain numerical solutions to solve such equations. The objectives of this research are to develop efficient and accurate numerical solutions in solving hyperbolic telegraph equations. Telegraph equations play an important role in modeling problems in physics, such as vibration of structures, transmission and propagation of electrical signals in a cable of transmission lines. Over the last decades, a great deal of attention has been raised up in the literature on the analysis, development and implementation of stable methods in solving one- and two-dimensional telegraph equations. In this study, a series of explicit group methods are proposed to solve two- and three-dimensional telegraph equations. The large sparse linear system that has resulted from the discretization is numerically solved by using the Gauss-Seidel relaxation iterative method, and the proposed methods are proven to be unconditionally stable by using the matrix method to analyze the convergence analysis. The parallel performance of all these explicit group methods were rigorously examined by utilizing the domain decomposition technique to divide the tasks involved in solving the equation and implementing parallelization under the Open specifications for Multi-Processing programming environment. By implementing this parallelization, computational costs of the solution process are reduced using multiple-core technologies. An ordering strategy is introduced to
the proposed methods in order to prevent any conflict on the usage of points among sub-domains. The parallelization algorithms were coded in C++ programming language using the Open specifications for Multi-Processing for parallel implementation libraries, and the experiments were performed using a sixteen-processor virtual machine. The findings and results from all these experiments show that the modified explicit decoupled group and explicit decoupled group methods are proven to be the most efficient in terms of execution timings for two- and three- dimensional telegraph equations respectively. The parallel performance shows the improvement of the execution times for the explicit group series.
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Keywords
New Parallel Iterative Methods , For The Solution Of Telegraph Equations