In this work, we study numerically two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equations. A centered finite difference method is exploited to discretize the spatial variables and leads to a system of the ordinary differential equation, in which the resulting coefficient matrix is complex symmetric and possesses the block Toeplitz structure. An exponential Runge–Kutta method is employed to solve such a system of the ordinary differential equation. Theoretically, the proposed method is second-order accuracy in space and fourth-order accuracy in time, respectively. In the practical implementation, the product of a block Toeplitz matrix exponential and a vector is calculated by the shift-invert Lanczos method. Meanwhile, the sectorial operator (the coefficient matrix) guarantees the fast approximation by the shift-invert Lanczos method. Numerical experiments are carried out to testify the theoretical results and demonstrate that the proposed method enjoys the excellent computational advantage.

在这项工作中，我们研究了二维非线性空间分数阶复数Ginzburg-Landau方程。利用中心有限差分法将空间变量离散化，得到一个常微分方程组，其中所得系数矩阵为复对称且具有块托普利兹结构。指数的Runge-Kutta方法用于求解这种常微分方程组。从理论上讲，该方法分别是空间的二阶精度和时间的四阶精度。在实际实现中，块托普利兹矩阵指数与矢量的乘积是通过移位反转兰科斯方法来计算的。与此同时，扇形算子（系数矩阵）通过移位反转Lanczos方法保证了快速逼近。数值实验验证了理论结果，证明了该方法具有良好的计算优势。