In this paper, a second-order linearized discontinuous Galerkin method on general meshes, which treats the backward differentiation formula of order two (BDF2) and Crank-Nicolson schemes as special cases, is proposed for solving the two-dimensional Ginzburg-Landau equations with cubic nonlinearity. By utilizing the discontinuous Galerkin inverse inequality and the mathematical induction method, the unconditionally optimal error estimate in $L^2$-norm is obtained. The core of the analysis in this paper resides in the classification and discussion of the relationship between the temporal step size and the spatial step size, specifically distinguishing between the two scenarios of tau^2 \leq h^{k+1}$and$\tau^2 > h^{k+1}$, where$k$denotes the degree of the discrete spatial scheme. Finally, this paper presents two numerical examples involving various grids and polynomial degrees to verify the correctness of the theoretical results.

翻译：本文针对具有立方非线性项的二维Ginzburg-Landau方程，提出了一种适用于一般网格的二阶线性化间断伽辽金方法，该方法将二阶后向差分公式与Crank-Nicolson格式作为特例包含其中。通过运用间断伽辽金逆不等式和数学归纳法，获得了$L^2$范数下的无条件最优误差估计。本文分析的核心在于对时间步长与空间步长关系的分类讨论，具体区分了$\tau^2 \leq h^{k+1}$和$\tau^2 > h^{k+1}$两种情形，其中$k$表示离散空间格式的多项式次数。最后，本文通过包含不同网格类型和多项式次数的两个数值算例验证了理论结果的正确性。