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.

翻译：采用线性隐式守恒差分格式对分数阶拉普拉斯吸引型耦合非线性薛定谔方程进行离散，可得到复对称线性系统，其系统矩阵为不定矩阵且呈托普利兹加对角结构。现有方法中既无高效的预条件迭代法，也无快速直接法能有效处理此类系统。本文基于复线性系统等效实块形式的正规分裂，提出一种新型矩阵分裂迭代法。该迭代法无条件收敛，并推导出准最优迭代参数。由此自然获得相应的新型预条件子，其构造简便且可通过快速傅里叶变换高效实现。理论分析表明，预条件后系统矩阵的特征值高度聚集。数值实验显示，该预条件子能显著加速Krylov子空间迭代法的收敛速度。特别地，相关预条件GMRES迭代法的收敛行为具有空间网格尺寸无关性，且几乎不受分数阶阶数影响。此外，结合预条件GMRES迭代法的线性隐式守恒差分格式能在给定精度下保持离散质量和能量守恒。