A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov $L2$-$1_{\sigma }$ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims.

中文翻译：

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非线性时空分数分数Ginzburg-Landau复杂系统的Alikhanov Legendre-Galerkin谱方法

提出并分析了时空分数耦合Ginzburg-Landau系统的有限差分/ Galerkin谱离散化。阿里汉诺夫$\mathrm{大号}2$--${\mathrm{1个}}_{\sigma }$差分公式用于离散时间Caputo分数阶导数，而Legendre-Galerkin谱近似用于近似Riesz空间分数算子。示出了该方案有效地适用于空间的频谱精度和时间的二阶。分数形式的Grönwall不等式的离散形式被用于建立基于离散能量估计技术的近似解的误差估计。数值延展实施的关键方面通过一些数值实验加以补充，以确认理论要求。