Volume 7, Issue 1
On Preconditioners Based on HSS for the Space Fractional CNLS Equations

Yu-Hong Ran, Jun-Gang Wang & Dong-Ling Wang

East Asian J. Appl. Math., 7 (2017), pp. 70-81.

Published online: 2018-02

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  • Abstract

The space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric Toeplitz-plusdiagonal matrix. In this paper, we present new preconditioners based on Hermitian and skew-Hermitian splitting (HSS) for such Toeplitz-like matrix. Theoretically, we show that all the eigenvalues of the resulting preconditioned matrices lie in the interior of the disk of radius 1 centered at the point (1, 0). Thus Krylov subspace methods with the proposed preconditioners converge very fast. Numerical examples are given to illustrate the effectiveness of the proposed preconditioners.

  • Keywords

The space fractional Schrödinger equations, Toeplitz matrix, Hermitian and skew-Hermitian splitting, preconditioner, Krylov subspace methods.

  • AMS Subject Headings

65F10, 65F15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-7-70, author = {}, title = {On Preconditioners Based on HSS for the Space Fractional CNLS Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {7}, number = {1}, pages = {70--81}, abstract = {

The space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric Toeplitz-plusdiagonal matrix. In this paper, we present new preconditioners based on Hermitian and skew-Hermitian splitting (HSS) for such Toeplitz-like matrix. Theoretically, we show that all the eigenvalues of the resulting preconditioned matrices lie in the interior of the disk of radius 1 centered at the point (1, 0). Thus Krylov subspace methods with the proposed preconditioners converge very fast. Numerical examples are given to illustrate the effectiveness of the proposed preconditioners.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.190716.051116b}, url = {http://global-sci.org/intro/article_detail/eajam/10735.html} }
TY - JOUR T1 - On Preconditioners Based on HSS for the Space Fractional CNLS Equations JO - East Asian Journal on Applied Mathematics VL - 1 SP - 70 EP - 81 PY - 2018 DA - 2018/02 SN - 7 DO - http://doi.org/10.4208/eajam.190716.051116b UR - https://global-sci.org/intro/article_detail/eajam/10735.html KW - The space fractional Schrödinger equations, Toeplitz matrix, Hermitian and skew-Hermitian splitting, preconditioner, Krylov subspace methods. AB -

The space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric Toeplitz-plusdiagonal matrix. In this paper, we present new preconditioners based on Hermitian and skew-Hermitian splitting (HSS) for such Toeplitz-like matrix. Theoretically, we show that all the eigenvalues of the resulting preconditioned matrices lie in the interior of the disk of radius 1 centered at the point (1, 0). Thus Krylov subspace methods with the proposed preconditioners converge very fast. Numerical examples are given to illustrate the effectiveness of the proposed preconditioners.

Yu-Hong Ran, Jun-Gang Wang & Dong-Ling Wang. (2020). On Preconditioners Based on HSS for the Space Fractional CNLS Equations. East Asian Journal on Applied Mathematics. 7 (1). 70-81. doi:10.4208/eajam.190716.051116b
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