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On structure preserving and circulant preconditioners for the space fractional coupled nonlinear Schrödinger equations
Author(s) -
Wang JunGang,
Ran YuHong,
Wang DongLing
Publication year - 2018
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2159
Subject(s) - preconditioner , toeplitz matrix , mathematics , circulant matrix , krylov subspace , eigenvalues and eigenvectors , discretization , coefficient matrix , biconjugate gradient stabilized method , matrix (chemical analysis) , identity matrix , linear system , mathematical analysis , pure mathematics , discrete mathematics , physics , materials science , quantum mechanics , composite material
Summary When the implicit, conservative difference scheme with the fractional centered difference formula is employed to discretize the space fractional coupled nonlinear Schrödinger equations, in each time step, we need to solve a complex symmetric linear system. The real part of the coefficient matrix is a symmetric Toeplitz‐plus‐diagonal matrix, whereas the imaginary part is the identity matrix. In this paper, a structure preserving preconditioner and a circulant preconditioner are proposed for such Toeplitz‐like matrix. Theoretically, tight bounds for eigenvalues of the preconditioned matrices are derived. Numerical implementations show that Krylov subspace iteration methods such as BiCGSTAB, when accelerated by the proposed preconditioners, are efficient solvers for solving the discretized linear system.