Applied and Computational Mathematics

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A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary

Received: Jul. 26, 2018    Accepted: Aug. 13, 2018    Published: Sep. 11, 2018
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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.

DOI 10.11648/j.acm.20180704.11
Published in Applied and Computational Mathematics ( Volume 7, Issue 4, August 2018 )
Page(s) 180-187
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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), 2024. Published by Science Publishing Group

Keywords

Helmholtz Equation, Fourier Transformation, Parallel Implementation

References
[1] Jin J M, Liu J, Z. Lou, et al. A fully high-order finite-element simulation of scattering by deep cavities [J]. Antennas & Propagation IEEE Transactions on, 2003 51 (9): 2420-2429.
[2] Li P J, and Wood A. A two-dimensional Helmholtz equation solution for the multiple cavity scattering problem [J]. Journal of Computational Physics, 2013 240 (1): 100-120.
[3] Liu J, Jin J M. A special higher order finite-element method for scattering by deep cavities [J]. IEEE Transactions on Antennas & Propagation, 2000 48 (5): 694-703.
[4] Wang Y X, Du K, Sun W W. A Second-Order Method for the Electromagnetic Scattering from a Large Cavity [J]. Numerical Mathematics: Theory, Methods and Applications,. 2008 1 (4): 357-382.
[5] Fu Y. Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers [J]. Journal of Computational Mathematics, 2008 26 (1): 98-111.
[6] Hicks G J. Arbitrary source and receiver positioning in finite-difference schemes using Kaiser windowed sinc functions [J]. Geophysics, 2002 67 (1): 156-166.
[7] Singer I, Turkel E. A perfectly matched layer for the Helmholtz equation in a semi-infinite strip [J]. Journal of Computational Physics, 2004 201 (1): 439-465.
[8] Britt S, Tsynkov S, Turkel E. A Compact Fourth Order Scheme for the Helmholtz Equation in Polar Coordinates [J]. Journal of Scientific Computing, 2010, 45 (1): 26-47.
[9] Nabavi M, Siddiqui M. H. K, Dargahi J. A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation [J]. Journal of Sound & Vibration, 2007 307(3): 972-982.
[10] Feng X F, Li Z L, Qiao Z H. High order compact finite difference schemes for the Helmholtz equation with discontinuous coefficients [J]. Journal of Computational Mathematics, 2011, 29 (3):324-340.
[11] Singer I, Turkel E. Sixth-order accurate finite difference schemes for the Helmholtz equation [J]. Journal of Computational Acoustics, 2006 14 (3): 339-351.
[12] Sutmann G. Compact finite difference schemes of sixth order for the Helmholtz equation [J]. Journal of Computational & Applied Mathematics, 2007 203 (1): 15-31.
[13] Sutmann G. Compact finite difference schemes of sixth order for the Helmholtz equation [J]. Journal of Computational & Applied Mathematics, 2007 203 (1): 15-31.
[14] Gordan D, Gordon R. Parallel solution of high frequency Helmholtz equations using high order finite difference schemes [J]. Applied Mathematics & Computation, 2012 218 (21): 10737-10754.
[15] Helmholtz solver by domain decomposition and modified Fourier method [J]. Siam Journal on Scientific Computing, 2005 26 (5): 1504-1524.
[16] Turkel E, Gordan D, Gordon R, et al.. Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number [J]. Journal of Computational Physics, 2013 232 (1): 72-287.
[17] Lu Y Y. A fourth-order Magnus scheme for Helmholtz equation [J]. Journal of Computational & Applied Mathematic, 2005 173 (2): 247-258.
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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

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    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

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    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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  • @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://download.sciencepg.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}
    }
    

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  • 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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Author Information
  • School of Mathematics and Physics, North China Electric Power University, Baoding, China

  • School of Mathematics and Physics, North China Electric Power University, Baoding, China

  • School of Mathematics and Physics, North China Electric Power University, Baoding, China

  • Section