Year 2020, Volume 49 , Issue 4, Pages 1245 - 1260 2020-08-06

A new double-step method for solving complex Helmholtz equation

Tahereh SALİMİ SİAHKOALAEİ [1] , Davod KHOJASTEH SALKUYEH [2]


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.

******************************************************************************

Complex Helmholtz equation, iterative method, complex linear systems, symmetric positive definite, double-step method
  • [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.
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0003-2121-2005
Author: Tahereh SALİMİ SİAHKOALAEİ
Institution: University of Guilan
Country: Iran


Orcid: 0000-0003-0228-8565
Author: Davod KHOJASTEH SALKUYEH (Primary Author)
Institution: University of Guilan
Country: Iran


Dates

Publication Date : August 6, 2020

Bibtex @research article { hujms494876, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1245 - 1260}, doi = {10.15672/hujms.494876}, title = {A new double-step method for solving complex Helmholtz equation}, key = {cite}, author = {Sali̇mi̇ Si̇ahkoalaei̇, Tahereh and Khojasteh Salkuyeh, Davod} }
APA Sali̇mi̇ Si̇ahkoalaei̇, 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 . DOI: 10.15672/hujms.494876
MLA Sali̇mi̇ Si̇ahkoalaei̇, T , Khojasteh Salkuyeh, D . "A new double-step method for solving complex Helmholtz equation" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1245-1260 <https://dergipark.org.tr/en/pub/hujms/issue/56305/494876>
Chicago Sali̇mi̇ Si̇ahkoalaei̇, T , Khojasteh Salkuyeh, D . "A new double-step method for solving complex Helmholtz equation". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1245-1260
RIS TY - JOUR T1 - A new double-step method for solving complex Helmholtz equation AU - Tahereh Sali̇mi̇ Si̇ahkoalaei̇ , Davod Khojasteh Salkuyeh Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.494876 DO - 10.15672/hujms.494876 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1245 EP - 1260 VL - 49 IS - 4 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.494876 UR - https://doi.org/10.15672/hujms.494876 Y2 - 2019 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics A new double-step method for solving complex Helmholtz equation %A Tahereh Sali̇mi̇ Si̇ahkoalaei̇ , Davod Khojasteh Salkuyeh %T A new double-step method for solving complex Helmholtz equation %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 4 %R doi: 10.15672/hujms.494876 %U 10.15672/hujms.494876
ISNAD Sali̇mi̇ Si̇ahkoalaei̇, 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
AMA Sali̇mi̇ Si̇ahkoalaei̇ T , Khojasteh Salkuyeh D . A new double-step method for solving complex Helmholtz equation. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1245-1260.
Vancouver Sali̇mi̇ Si̇ahkoalaei̇ T , Khojasteh Salkuyeh D . A new double-step method for solving complex Helmholtz equation. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1245-1260.