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On Preconditioners Based on HSS for the Space Fractional CNLS Equations

Part of: Numerical linear algebra

Published online by Cambridge University Press:  31 January 2017

Yu-Hong Ran ,
Jun-Gang Wang  and
Dong-Ling Wang
Yu-Hong Ran*
Affiliation:
Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi'an, Shaanxi 710127, China
Jun-Gang Wang
Affiliation:
Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China
Dong-Ling Wang
Affiliation:
Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi'an, Shaanxi 710127, China
*
*Corresponding author. Email address:ranyh@nwu.edu.cn (Y.-H. Ran)
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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 equationsToeplitz matrixHermitian and skew-Hermitian splittingpreconditionerKrylov subspace methods

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F15: Eigenvalues, eigenvectors
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 1 , February 2017 , pp. 70 - 81
Copyright
Copyright © Global-Science Press 2017 

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