Space and time approximations for two-dimensional space fractional complex Ginzburg–Landau equation are examined. The schemes under consideration are discreted by the second-order backward differential formula (BDF2) in time and two classes of the fractional centered finite difference methods in space. A linearized technique is employed by the extrapolation. We prove the unique solvability and stability for both numerical methods. The convergence of both numerical methods is analyzed at length utilizing the energy argument, and the convergence orders under the optimal step size ratio are $O({\tau }^2+h^2)$ and $O({\tau }^2+h^4)$ in the sense of the discrete $L^2$-norm, where $\tau$ is the time step size, $h=max\{h_x,h_y\}$, and $h_x,h_y$ are spatial grid sizes in the $x$-direction and $y$-direction, respectively. In addition, we construct a multistep alternating direction implicit (ADI) scheme and a multistep compact ADI scheme based on BDF2 for the efficiently numerical implementation. Finally, numerical examples are carried out to verify our theoretical results.

研究了二维空间分数复数Ginzburg-Landau方程的时空近似。所考虑的方案在时间上由二阶后向微分公式（BDF2）和空间中的两类分数中心有限差分法分开。外推采用线性化技术。我们证明了这两种数值方法都具有独特的可溶性和稳定性。利用能量参数对两种数值方法的收敛性进行了详细分析，最优步长比下的收敛阶为$Ø（{\tau }^2+H^2）$ 和 $Ø（{\tau }^2+H^4）$ 在离散意义上 ${\mathrm{大号}}^2$-规范，在哪里 $\tau$ 是时间步长， $H=最高\{H_X，H_{ÿ}\}$和 $H_X，H_{ÿ}$ 是空间网格的大小 $X$方向和 $ÿ$方向。此外，为了高​​效地实现数值计算，我们基于BDF2构造了多步交替方向隐式（ADI）方案和多步紧凑ADI方案。最后，通过数值算例验证了我们的理论结果。