Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T19:33:48.331Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bai, Z.-Z., Benzi, M., Chen, F., Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87(2010), pp. 93–111.Google Scholar
[2] Bai, Z.-Z., Benzi, M., Chen, F., On preconditioned MHSS iteration methods for complex symmetric linear systems, Numerical Algorithms, 56(2011), pp. 297–317.Google Scholar
[3] Bai, Z.-Z., Golub, G.H. and Li, C.-K., Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Mathematics of Computation, 76(2007), pp. 287–298.Google Scholar
[4] Bai, Z.-Z., Golub, G.H., Ng, M.K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM Journal on Matrix Analysis and Applications, 24(2003), pp. 603–626.Google Scholar
[5] Bai, Z.-Z., Golub, G.H., Ng, M.K., On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra and its Applications, 428(2008), pp. 413–440.Google Scholar
[6] Bai, Z.-Z., Golub, G.H., Pan, J.-Y., Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numerische Mathematik, 98(2004), pp. 1–32.Google Scholar
[7] Chan, R.H., Jin, X.-Q., An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, 2007.Google Scholar
[8] Chan, R.H., Ng, M.K., Conjugate gradient methods for Toeplitz systems, SIAM Review, 38(1996), pp. 427–482.Google Scholar
[9] Chan, R.H., Ng, K.P., Fast iterative solvers for Toeplitz-plus-band systems, SIAM Journal on Scientific Computing, 14(1993), pp. 1013–1019.Google Scholar
[10] Chan, R.H., Strang, G., Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM Journal on Scientific and Statistical Computing, 10(1989), pp. 104–119.Google Scholar
[11] Demengel, F., Demengel, G., Fractional Sobolev Spaces, Springer, London, 2012.Google Scholar
[12] Laskin, N., Fractional quantum mechanics and Lévy path integrals, Physics Letters A, 268(2000), pp. 298–305.Google Scholar
[13] Laskin, N., Fractional Schrödinger equation, Physical Review E, 66(2002), pp. 56–108.Google Scholar
[14] Lei, S.-L., Sun, H.-W., A circulant preconditioner for fractional diffusion equations, Journal of Computational Physics, 242(2013), pp. 715–725.Google Scholar
[15] Ng, M.K., Iterative Methods for Toeplitz Systems, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004.Google Scholar
[16] Ng, M.K., Pan, J.-Y., Approximate inverse circulant-plus-diagonal preconditioners for Toeplitz-plus-diagonal matrices, SIAM Journal on Scientific Computing, 32(2010), pp. 1442–1464.Google Scholar
[17] Ng, M.K., Serra-Capizzano, S., Tablino-Possio, C., Multigrid methods for symmetric Sinc-Galerkin systems, Linear Algebra and its Applications, 12(2005), pp. 261–269.Google Scholar
[18] Ortigueira, M.D., Riesz potential opeators and inverses via fractional centred derivatives, International Journal of Mathematics and Mathematical Sciences, 2006(2006), pp. 1–12.Google Scholar
[19] Pan, J.-Y., Ke, R.-H., Ng, M. K., Sun, H.-W., Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations, SIAM Journal on Scientific Computing, 36(2014), pp. A2698–A2719.Google Scholar
[20] Ran, Y.-H., Wang, J.-G., Wang, D.-L., On HSS-like iteration method for the space fractional coupled nonlinear Schrödinger equations, Applied Mathematics and Computation, 271(2015), pp. 482–488.Google Scholar
[21] Wang, D.-L., Xiao, A.-G., Yang, W., Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative, Journal of Computational Physics, 242(2013), pp. 670–681.Google Scholar
[22] Wang, D.-L., Xiao, A.-G., Yang, W., A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations, Journal of Computational Physics, 272(2014), pp. 644–655.Google Scholar
[23] Wang, D.-L., Xiao, A.-G., Yang, W., Maximum-norm error analysis of a difference scheme for the space fractional CNLS, Applied Mathematics and Computation, 257(2015), pp. 241–251.Google Scholar
[24] Yang, Q., Liu, F., Turner, I., Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Applied Mathmatical Modelling, 34(2010), pp. 200–218.Google Scholar