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 behavior of the corresponding preconditioned GMRES method exhibits a linear dependence on the space mesh size, which weakens as the fractional order parameter decreases.

翻译：通过应用[D.-L. Wang, A.-G. Xiao, W. Yang. J. Comput. Phys. 2014;272:670-681]中提出的线性隐式守恒差分格式，具有排斥性的空间分数阶耦合非线性薛定谔方程系统导出了一系列具有复对称及Toeplitz加对角结构的线性方程组。本文提出了一种具有Toeplitz块分裂的斜对角与正规迭代方法来求解上述线性方程组。该新迭代方法被证明是无条件收敛的，并推导出了最优迭代参数。自然地，该新迭代方法导出了一个具有循环块结构的斜对角与正规预条件子，可通过快速算法高效执行。理论上，我们给出了离散分数阶拉普拉斯算子及其循环逼近的特征值的尖锐界，进一步分析表明预条件后系统矩阵的谱分布是紧致的。数值实验表明，新预条件子能显著提高Krylov子空间迭代方法的计算效率。此外，相应的预条件GMRES方法的表现呈现出对空间网格尺寸的线性依赖性，且该依赖性随着分数阶参数的减小而减弱。