TY - JOUR
T1 - On Preconditioners Based on HSS for the Space Fractional CNLS Equations
JO - East Asian Journal on Applied Mathematics
VL - 1
SP - 70
EP - 81
PY - 2018
DA - 2018/02
SN - 7
DO - http://doi.org/10.4208/eajam.190716.051116b
UR - https://global-sci.org/intro/article_detail/eajam/10735.html
KW - The space fractional Schrödinger equations, Toeplitz matrix, Hermitian and skew-Hermitian splitting, preconditioner, Krylov subspace methods.
AB - <p style="text-align: justify;">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-plus-diagonal
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.</p>