Fast iterative solvers and simulation for the space fractional Ginzburg–Landau equations

In this paper, we propose a new splitting iteration method for solving the Toeplitz-like complex linear system arising from the space fractional Ginzburg–Landau equations, of which the coefficient matrix is equal to the sum of a complex scaled Toeplitz-plus-diagonal matrix and a symmetric Toeplitz-plus-diagonal matrix. The new splitting method enjoys computational advantage since circulant preconditioner and fast Fourier transform(FFT) can be used for solving the involved linear subsystem. Under some conditions, the convergence properties of the corresponding iteration method are derived. Numerical examples are given to illustrate the effectiveness of the proposed method.