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.
| Published in | Applied and Computational Mathematics (Volume 7, Issue 4) |
| DOI | 10.11648/j.acm.20180704.11 |
| Page(s) | 180-187 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2018. Published by Science Publishing Group |
Helmholtz Equation, Fourier Transformation, Parallel Implementation
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APA Style
Sheng An, Gendai Gu, Meiling Zhao. (2018). A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary. Applied and Computational Mathematics, 7(4), 180-187. https://doi.org/10.11648/j.acm.20180704.11
ACS Style
Sheng An; Gendai Gu; Meiling Zhao. A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary. Appl. Comput. Math. 2018, 7(4), 180-187. doi: 10.11648/j.acm.20180704.11
AMA Style
Sheng An, Gendai Gu, Meiling Zhao. A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary. Appl Comput Math. 2018;7(4):180-187. doi: 10.11648/j.acm.20180704.11
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Download@article{10.11648/j.acm.20180704.11,
author = {Sheng An and Gendai Gu and Meiling Zhao},
title = {A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary},
journal = {Applied and Computational Mathematics},
volume = {7},
number = {4},
pages = {180-187},
doi = {10.11648/j.acm.20180704.11},
url = {https://doi.org/10.11648/j.acm.20180704.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180704.11},
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.},
year = {2018}
}
TY - JOUR T1 - A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary AU - Sheng An AU - Gendai Gu AU - Meiling Zhao Y1 - 2018/09/11 PY - 2018 N1 - https://doi.org/10.11648/j.acm.20180704.11 DO - 10.11648/j.acm.20180704.11 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 180 EP - 187 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20180704.11 AB - 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. VL - 7 IS - 4 ER -
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