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.

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