A FAST COMPACT DIFFERENCE METHOD FOR TWO-DIMENSIONAL NONLINEAR SPACE-FRACTIONAL COMPLEX GINZBURG-LANDAU EQUATIONS

This paper focuses on a fast and high-order finite difference method for two-dimensional space-fractional complex Ginzburg-Landau equations.We firstly establish a three-level finite difference scheme for the time variable,followed by the linearized technique of the nonlinear term.Then the fourth-order compact finite difference method is employed to discretize the spatial variables.Hence the accuracy of the discretization is O(τ^(2)+h^(4)_(1)+h^(4)_(2))in L_(2)-norm,where τ is the temporal step-size,both h_(1) and h_(2) denote spatial mesh sizes in x-and y-directions,respectively.The rigorous theoretical analysis,including the uniqueness,the almost unconditional stability,and the convergence,is studied via the energy argument.Practically,the discretized system holds the block Toeplitz structure.Therefore,the coefficient Toeplitz-like matrix only requires O(M_(1)M_(2)) memory storage,and the matrix-vector multiplication can be carried out in O(M_(1)M_(2))(log M_(1)+log M_(2))computational complexity by the fast Fourier transformation,where M_(1) and M_(2) denote the numbers of the spatial grids in two different directions.In order to solve the resulting Toeplitz-like system quickly,an efficient preconditioner with the Krylov subspace method is proposed to speed up the iteration rate.Numerical results are given to demonstrate the well performance of the proposed method.