A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary
Applied and Computational Mathematics
Volume 7, Issue 4, August 2018, Pages: 180-187
Received: Jul. 26, 2018; Accepted: Aug. 13, 2018; Published: Sep. 11, 2018
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Authors
Sheng An, School of Mathematics and Physics, North China Electric Power University, Baoding, China
Gendai Gu, School of Mathematics and Physics, North China Electric Power University, Baoding, China
Meiling Zhao, School of Mathematics and Physics, North China Electric Power University, Baoding, China
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Abstract
Helmholtz equation is widely applied in the scientific and engineering problem. For the solution of the three-dimensional Helmholtz equation, the computational efficiency of the algorithm is especially important. In this paper, in order to solve the contradiction between accuracy and efficiency, a fast high order finite difference method is proposed for solving the three-dimensional Helmholtz equation. First, a traditional fourth order method is constructed. Then, fast Fourier transformation are introduced to generate a block-tridiagonal structure which can easily divide the original problem into small and independent subsystems. For large 3D problems, the computation of traditional discrete Fourier transformation is less efficient, and the memory requirements increase rapidly. To fix this problem, the transformed coefficient matrix is constructed as a sparse structure. In light of the sparsity, the algorithm presented in this paper requires less memory space and computational cost. This sparse structure also leads to independent solving procedure of any plane in the domain. Therefore, parallel method can be used to solve the Helmholtz equation with large grid number. In the end, three numerical experiments are presented to verify the effectiveness of the fast fourth-order algorithm, and the acceleration effect to use the parallel method has been demonstrated.
Keywords
Helmholtz Equation, Fourier Transformation, Parallel Implementation
To cite this article
Sheng An, Gendai Gu, Meiling Zhao, A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary, Applied and Computational Mathematics. Vol. 7, No. 4, 2018, pp. 180-187. doi: 10.11648/j.acm.20180704.11
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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