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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], thesystem of repulsive space fractional coupled nonlinear Schr\"odinger equationsleads to a sequence of linear systems with complex symmetric andToeplitz-plus-diagonal structure. In this paper, we propose the diagonal andnormal with Toeplitz-block splitting iteration method to solve the above linearsystems. The new iteration method is proved to converge unconditionally, andthe optimal iteration parameter is deducted. Naturally, this new iterationmethod leads to a diagonal and normal with circulant-block preconditioner whichcan be executed efficiently by fast algorithms. In theory, we provide sharpbounds for the eigenvalues of the discrete fractional Laplacian and itscirculant approximation, and further analysis indicates that the spectraldistribution of the preconditioned system matrix is tight. Numericalexperiments show that the new preconditioner can significantly improve thecomputational efficiency of the Krylov subspace iteration methods. Moreover,the corresponding preconditioned GMRES method shows space mesh size independentand almost fractional order parameter insensitive convergence behaviors.

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