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A new double-step method for solving complex Helmholtz equation

Year 2020, Volume: 49 Issue: 4, 1245 - 1260, 06.08.2020
https://doi.org/10.15672/hujms.494876

Abstract

We present a new double-step iteration method for solving the systems of linear equations that arise from finite difference discretizations of the complex Helmholtz equations. Convergence analysis of the method is discussed. An upper bound on the spectral radius of the iteration matrix of the method is presented and the parameter which minimizes this upper bound is computed. The proposed method is compared theoretically and numerically with some existing methods.

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References

  • [1] L. Abrahamsson, H.-O. Kreiss, Numerical solution of the coupled mode equations in duct acoustics, J. Comput. Phys. 111, 1–14, 1994.
  • [2] O. Axelsson, A. Kucherov, Real valued iterative methods for solving complex symmetric linear systems, Numer. Linear Algebra Appl. 7, 197–218, 2000.
  • [3] Z.-Z. Bai, M. Benzi, F. Chen, Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87, 93–111, 2010.
  • [4] Z.-Z. Bai, M. Benzi, F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer. Algor. 56, 297–317, 2011.
  • [5] Z.-Z. Bai, M. Benzi, F. Chen, Z.-Q. Wang, Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems, IMA J. Numer. Anal. 33, 343–369, 2013.
  • [6] Z.-Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24, 603–626, 2003.
  • [7] Z.-Z. Bai, G.H. Golub, M.K. Ng, On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl. 428, 413–440, 2008.
  • [8] Z.-Z. Bai, B.N. Parlett, Z.-Q.Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102, 1–38, 2005.
  • [9] M. Benzi, D. Bertaccini, Block preconditioning of real-valued iterative algorithms for complex linear systems, IMA J. Numer. Anal. 28, 598–618, 2008.
  • [10] O.G. Ernst, Fast numerical solution of Exterior Helmholtz with radiation boundary condition by imbedding, Ph.D thesis, Dept. of Computer Science, Stanford Univ., Stanford, CA, 1994.
  • [11] M.R. Hestenes, E.L. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Natl. Stand, Sec. B 49, 409–436, 1952.
  • [12] D. Hezari, V. Edalatpour, D.K. Salkuyeh, Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations, Numer. Linear Algebra Appl. 22, 761–776, 2015.
  • [13] D. Hezari, D.K. Salkuyeh, V. Edalatpour, A new iterative method for solving a class of complex symmetric system of linear equations, Numer. Algor. 73, 927–955, 2016.
  • [14] C.D. Meyer, Matrix analysis and applied linear algebra, SIAM, Philadelphia, 2000.
  • [15] Y. Saad, Iterative methods for sparse linear systems, PWS Press, New York, 1995.
  • [16] D.K. Salkuyeh, Two-step scale-splitting method for solving complex symmetric system of linear equations, arXiv:1705.02468.
  • [17] D.K. Salkuyeh, D. Hezari, V. Edalatpour, Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations, Int. J. Comput. Math. 92, 802–815, 2015.
  • [18] D.K. Salkuyeh, T.S. Siahkolaei, Two-parameter TSCSP method for solving complex symmetric system of linear equations, Calcolo 55, 8, 2018.
  • [19] T. Wang, Q. Zheng, L. Lu, A new iteration method for a class of complex symmetric linear systems, J. Comput. Appl. Math. 325, 188–197, 2017.
  • [20] Z. Zheng, F.-L. Huang, Y.-C. Peng, Double-step scale splitting iteration method for a class of complex symmetric linear systems, Appl. Math. Lett. 73, 91–97, 2017.
Year 2020, Volume: 49 Issue: 4, 1245 - 1260, 06.08.2020
https://doi.org/10.15672/hujms.494876

Abstract

References

  • [1] L. Abrahamsson, H.-O. Kreiss, Numerical solution of the coupled mode equations in duct acoustics, J. Comput. Phys. 111, 1–14, 1994.
  • [2] O. Axelsson, A. Kucherov, Real valued iterative methods for solving complex symmetric linear systems, Numer. Linear Algebra Appl. 7, 197–218, 2000.
  • [3] Z.-Z. Bai, M. Benzi, F. Chen, Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87, 93–111, 2010.
  • [4] Z.-Z. Bai, M. Benzi, F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer. Algor. 56, 297–317, 2011.
  • [5] Z.-Z. Bai, M. Benzi, F. Chen, Z.-Q. Wang, Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems, IMA J. Numer. Anal. 33, 343–369, 2013.
  • [6] Z.-Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24, 603–626, 2003.
  • [7] Z.-Z. Bai, G.H. Golub, M.K. Ng, On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl. 428, 413–440, 2008.
  • [8] Z.-Z. Bai, B.N. Parlett, Z.-Q.Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102, 1–38, 2005.
  • [9] M. Benzi, D. Bertaccini, Block preconditioning of real-valued iterative algorithms for complex linear systems, IMA J. Numer. Anal. 28, 598–618, 2008.
  • [10] O.G. Ernst, Fast numerical solution of Exterior Helmholtz with radiation boundary condition by imbedding, Ph.D thesis, Dept. of Computer Science, Stanford Univ., Stanford, CA, 1994.
  • [11] M.R. Hestenes, E.L. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Natl. Stand, Sec. B 49, 409–436, 1952.
  • [12] D. Hezari, V. Edalatpour, D.K. Salkuyeh, Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations, Numer. Linear Algebra Appl. 22, 761–776, 2015.
  • [13] D. Hezari, D.K. Salkuyeh, V. Edalatpour, A new iterative method for solving a class of complex symmetric system of linear equations, Numer. Algor. 73, 927–955, 2016.
  • [14] C.D. Meyer, Matrix analysis and applied linear algebra, SIAM, Philadelphia, 2000.
  • [15] Y. Saad, Iterative methods for sparse linear systems, PWS Press, New York, 1995.
  • [16] D.K. Salkuyeh, Two-step scale-splitting method for solving complex symmetric system of linear equations, arXiv:1705.02468.
  • [17] D.K. Salkuyeh, D. Hezari, V. Edalatpour, Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations, Int. J. Comput. Math. 92, 802–815, 2015.
  • [18] D.K. Salkuyeh, T.S. Siahkolaei, Two-parameter TSCSP method for solving complex symmetric system of linear equations, Calcolo 55, 8, 2018.
  • [19] T. Wang, Q. Zheng, L. Lu, A new iteration method for a class of complex symmetric linear systems, J. Comput. Appl. Math. 325, 188–197, 2017.
  • [20] Z. Zheng, F.-L. Huang, Y.-C. Peng, Double-step scale splitting iteration method for a class of complex symmetric linear systems, Appl. Math. Lett. 73, 91–97, 2017.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Tahereh Salimi Siahkoalaei This is me 0000-0003-2121-2005

Davod Khojasteh Salkuyeh 0000-0003-0228-8565

Publication Date August 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 4

Cite

APA Salimi Siahkoalaei, T., & Khojasteh Salkuyeh, D. (2020). A new double-step method for solving complex Helmholtz equation. Hacettepe Journal of Mathematics and Statistics, 49(4), 1245-1260. https://doi.org/10.15672/hujms.494876
AMA Salimi Siahkoalaei T, Khojasteh Salkuyeh D. A new double-step method for solving complex Helmholtz equation. Hacettepe Journal of Mathematics and Statistics. August 2020;49(4):1245-1260. doi:10.15672/hujms.494876
Chicago Salimi Siahkoalaei, Tahereh, and Davod Khojasteh Salkuyeh. “A New Double-Step Method for Solving Complex Helmholtz Equation”. Hacettepe Journal of Mathematics and Statistics 49, no. 4 (August 2020): 1245-60. https://doi.org/10.15672/hujms.494876.
EndNote Salimi Siahkoalaei T, Khojasteh Salkuyeh D (August 1, 2020) A new double-step method for solving complex Helmholtz equation. Hacettepe Journal of Mathematics and Statistics 49 4 1245–1260.
IEEE T. Salimi Siahkoalaei and D. Khojasteh Salkuyeh, “A new double-step method for solving complex Helmholtz equation”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, pp. 1245–1260, 2020, doi: 10.15672/hujms.494876.
ISNAD Salimi Siahkoalaei, Tahereh - Khojasteh Salkuyeh, Davod. “A New Double-Step Method for Solving Complex Helmholtz Equation”. Hacettepe Journal of Mathematics and Statistics 49/4 (August 2020), 1245-1260. https://doi.org/10.15672/hujms.494876.
JAMA Salimi Siahkoalaei T, Khojasteh Salkuyeh D. A new double-step method for solving complex Helmholtz equation. Hacettepe Journal of Mathematics and Statistics. 2020;49:1245–1260.
MLA Salimi Siahkoalaei, Tahereh and Davod Khojasteh Salkuyeh. “A New Double-Step Method for Solving Complex Helmholtz Equation”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, 2020, pp. 1245-60, doi:10.15672/hujms.494876.
Vancouver Salimi Siahkoalaei T, Khojasteh Salkuyeh D. A new double-step method for solving complex Helmholtz equation. Hacettepe Journal of Mathematics and Statistics. 2020;49(4):1245-60.