Fast Normal Splitting Preconditioner for Attractive Coupled Nonlinear Schrödinger Equations with a Fractional Laplacian

A linearly implicit conservative difference scheme is applied to discretize the attractive coupled nonlinear Schrödinger equations with a fractional Laplacian. In this case complex symmetric linear systems appear, with indefinite and Toeplitz-plus-diagonal system matrices. Standard fast methods of direct solution or iteration using a preconditioner are not applicable for such systems. A novel iterative method is proposed, based on the normal splitting of the equivalent real block form of linear systems. Unconditional convergence is proved and the quasi-optimal iteration parameter is deducted. The preconditioner for this method is obtained naturally; it is constructed and efficiently implemented using the fast Fourier transform. The theoretical analysis shows that the eigenvalues of the preconditioned system matrix are closely clustered. The numerical experiments demonstrate that the new preconditioner significantly speeds up the convergence rate of iterative Krylov subspace methods. In particular, the convergence behavior of the corresponding preconditioned generalized minimum residual method is independent of the mesh size and practically insensitive to the fractional order. Moreover, the linearly implicit conservative difference scheme in this case preserves the mass and energy with the given accuracy.