On TDS-PCG Iteration Method with Circulant Preconditioners for Solving the Space Fractional Coupled Nonlinear Schrödinger Equations.

The goal of this paper is to solve the complex symmetric linear systems generated from the discretization of the space fractional coupled nonlinear Schrödinger (CNLS) equations, whose coefficient matrix is equal to the sum of a symmetric positive definite Toeplitz matrix and a Hermitian positive definite complex diagonal matrix. In order to make the best use of the full Toeplitz structure of the coefficient matrix, a new Toeplitz and diagonal splitting (TDS) is given and the corresponding TDS iteration method is proposed to solve the discretized linear systems, then two circulant preconditioners based on Strang’s and T. Chan’s circulant approximation, are proposed to accelerate the convergence of the preconditioned conjugated gradient (PCG) method for solving the first linear sub-system in the TDS method. Theoretical analysis and numerical experiments demonstrate that the TDS method is unconditional convergence and the TDS-PCG inner-outer iteration method with two circulant preconditioners to solve the discretization linear systems of the space fractional CNLS equations is feasible and efficient. [ABSTRACT FROM AUTHOR]