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On Preconditioners Based on HSS for the Space Fractional CNLS Equations

  • Yu-Hong Ran (a1), Jun-Gang Wang (a2) and Dong-Ling Wang (a1)
Abstract
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.

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Corresponding author
*Corresponding author. Email address: ranyh@nwu.edu.cn (Y.-H. Ran)
References
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[1] Bai Z.-Z., Benzi M., Chen F., Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87(2010), pp. 93111.
[2] Bai Z.-Z., Benzi M., Chen F., On preconditioned MHSS iteration methods for complex symmetric linear systems, Numerical Algorithms, 56(2011), pp. 297317.
[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. 287298.
[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. 603626.
[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. 413440.
[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. 132.
[7] Chan R.H., Jin X.-Q., An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, 2007.
[8] Chan R.H., Ng M.K., Conjugate gradient methods for Toeplitz systems, SIAM Review, 38(1996), pp. 427482.
[9] Chan R.H., Ng K.P., Fast iterative solvers for Toeplitz-plus-band systems, SIAM Journal on Scientific Computing, 14(1993), pp. 10131019.
[10] Chan R.H., Strang G., Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM Journal on Scientific and Statistical Computing, 10(1989), pp. 104119.
[11] Demengel F., Demengel G., Fractional Sobolev Spaces, Springer, London, 2012.
[12] Laskin N., Fractional quantum mechanics and Lévy path integrals, Physics Letters A, 268(2000), pp. 298305.
[13] Laskin N., Fractional Schrödinger equation, Physical Review E, 66(2002), pp. 56108.
[14] Lei S.-L., Sun H.-W., A circulant preconditioner for fractional diffusion equations, Journal of Computational Physics, 242(2013), pp. 715725.
[15] Ng M.K., Iterative Methods for Toeplitz Systems, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004.
[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. 14421464.
[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. 261269.
[18] Ortigueira M.D., Riesz potential opeators and inverses via fractional centred derivatives, International Journal of Mathematics and Mathematical Sciences, 2006(2006), pp. 112.
[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. A2698A2719.
[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. 482488.
[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. 670681.
[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. 644655.
[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. 241251.
[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. 200218.
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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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