In this work, we propose an alternating direction implicit (ADI) scheme to discrete the two-dimensional spatial fractional Ginzburg–Landau equations. Meanwhile, a matrix splitting iteration method that preserves the Toeplitz structure is proposed for solving the resulting complex linear system, in which the circulant preconditioning technique and fast Fourier transform (FFT) can be adopted to improve computing efficiency. Convergence properties of the corresponding method are derived under some conditions. Numerical experiments demonstrate that our method outperforms some existing iteration methods in iterative steps and computing time.

中文翻译：

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二维空间分数Ginzburg-Landau方程的快速迭代求解器

在这项工作中，我们提出了一种交替方向隐式（ADI）方案来离散二维空间分数分数Ginzburg-Landau方程。同时，提出了一种保留了Toeplitz结构的矩阵分裂迭代方法来解决由此产生的复杂线性系统，其中可以采用循环预处理和快速傅立叶变换（FFT）来提高计算效率。在某些条件下得出相应方法的收敛性质。数值实验表明，我们的方法在迭代步骤和计算时间上均优于某些现有的迭代方法。