Preconditioned modified Hermitian and skew-Hermitian splitting iteration methods for fractional nonlinear Schrödinger equations

Abstract A variant of preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method is proposed for a class of Toeplitz-like complex linear equations arising from the space fractional coupled nonlinear Schrodinger equations. Theoretical analysis show that the new method is convergent unconditionally. Moreover, the choice of the theoretical optimal parameter is further studied in detail, which minimize the upper bound of the spectral radius and relate to the extreme eigenvalues of the Toeplitz matrix and the extreme absolute values of the diagonal matrix. In addition, the splitting can be used as a preconditioner for Krylov subspaces method. Numerical experiments show that the proposed method is efficient, and the optimal parameter is independent of the mesh size which is very useful in large sparse practical problem.

[1]  G. Strang,et al.  Toeplitz equations by conjugate gradients with circulant preconditioner , 1989 .

[2]  Jie Xin,et al.  Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation , 2008, Appl. Math. Comput..

[3]  Gene H. Golub,et al.  Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[4]  Yu-Hong Ran,et al.  On HSS-like iteration method for the space fractional coupled nonlinear Schrödinger equations , 2015, Appl. Math. Comput..

[5]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[6]  Xiaoyi Guo,et al.  Some physical applications of fractional Schrödinger equation , 2006 .

[7]  Ludwig Gauckler,et al.  Nonlinear Schrödinger Equations and Their Spectral Semi-Discretizations Over Long Times , 2010, Found. Comput. Math..

[8]  Siu-Long Lei,et al.  A circulant preconditioner for fractional diffusion equations , 2013, J. Comput. Phys..

[9]  Bülent Karasözen,et al.  Multi-symplectic integration of coupled non-linear Schrödinger system with soliton solutions , 2009, Int. J. Comput. Math..

[10]  Wei Yang,et al.  Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative , 2013, J. Comput. Phys..

[11]  Jungang Wang,et al.  On Preconditioners Based on HSS for the Space Fractional CNLS Equations , 2017 .

[12]  Jie Xin,et al.  The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition , 2011, Comput. Math. Appl..

[13]  Michael K. Ng,et al.  Preconditioning Techniques for Diagonal-times-Toeplitz Matrices in Fractional Diffusion Equations , 2014, SIAM J. Sci. Comput..

[14]  Dongling Wang,et al.  On partially inexact HSS iteration methods for the complex symmetric linear systems in space fractional CNLS equations , 2017, J. Comput. Appl. Math..

[15]  Zhong-Zhi Bai,et al.  On preconditioned iteration methods for complex linear systems , 2015 .

[16]  Wei Yang,et al.  A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations , 2014, J. Comput. Phys..

[17]  T. Chan.  An Optimal Circulant Preconditioner for Toeplitz Systems , 1988 .

[18]  Jian-Qiang Sun,et al.  Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system , 2003 .

[19]  M. Benzi,et al.  Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems , 2013 .

[20]  R. Chan,et al.  An Introduction to Iterative Toeplitz Solvers , 2007 .

[21]  N. Laskin.  Fractional quantum mechanics and Lévy path integrals , 1999, hep-ph/9910419.

[22]  Jianyu Pan,et al.  Approximate Inverse Circulant-plus-Diagonal Preconditioners for Toeplitz-plus-Diagonal Matrices , 2010, SIAM J. Sci. Comput..

[23]  Zhi-Zhong Sun,et al.  On the L∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations , 2010, Comput. Math. Appl..

[24]  Wei Yang,et al.  Maximum-norm error analysis of a difference scheme for the space fractional CNLS , 2015, Appl. Math. Comput..