Diagonal and normal with Toeplitz-block splitting iteration method for space fractional coupled nonlinear Schr\"odinger equations with repulsive nonlinearities

By applying the linearly implicit conservative difference scheme proposed in [D.-L. Wang, A.-G. Xiao, W. Yang. J. Comput. Phys. 2014;272:670-681], the system of repulsive space fractional coupled nonlinear Schr\"odinger equations leads to a sequence of linear systems with complex symmetric and Toeplitz-plus-diagonal structure. In this paper, we propose the diagonal and normal with Toeplitz-block splitting iteration method to solve the above linear systems. The new iteration method is proved to converge unconditionally, and the optimal iteration parameter is deducted. Naturally, this new iteration method leads to a diagonal and normal with circulant-block preconditioner which can be executed efficiently by fast algorithms. In theory, we provide sharp bounds for the eigenvalues of the discrete fractional Laplacian and its circulant approximation, and further analysis indicates that the spectral distribution of the preconditioned system matrix is tight. Numerical experiments show that the new preconditioner can significantly improve the computational efficiency of the Krylov subspace iteration methods. Moreover, the corresponding preconditioned GMRES method shows space mesh size independent and almost fractional order parameter insensitive convergence behaviors

A fast normal splitting preconditioner for attractive coupled nonlinear Schr\"odinger equations with fractional Laplacian

A linearly implicit conservative difference scheme is applied to discretize the attractive coupled nonlinear Schr\"odinger equations with fractional Laplacian. Complex symmetric linear systems can be obtained, and the system matrices are indefinite and Toeplitz-plus-diagonal. Neither efficient preconditioned iteration method nor fast direct method is available to deal with these systems. In this paper, we propose a novel matrix splitting iteration method based on a normal splitting of an equivalent real block form of the complex linear systems. This new iteration method converges unconditionally, and the quasi-optimal iteration parameter is deducted. The corresponding new preconditioner is obtained naturally, which can be constructed easily and implemented efficiently by fast Fourier transform. Theoretical analysis indicates that the eigenvalues of the preconditioned system matrix are tightly clustered. Numerical experiments show that the new preconditioner can significantly accelerate the convergence rate of the Krylov subspace iteration methods. Specifically, the convergence behavior of the related preconditioned GMRES iteration method is spacial mesh-size-independent, and almost fractional order insensitive. Moreover, the linearly implicit conservative difference scheme in conjunction with the preconditioned GMRES iteration method conserves the discrete mass and energy in terms of a given precision