This paper deals with the numerical simulation of the 2D magnetic time-dependent Ginzburg-Landau (TDGL) equations in the regime of small but finite (inverse) Ginzburg-Landau parameter $\epsilon$ and constant (order $1$ in $\epsilon$) applied magnetic field. In this regime, a well-known feature of the TDGL equation is the appearance of quantized vortices with core size of order $\epsilon$. Moreover, in the singular limit $\epsilon \searrow 0$, these vortices evolve according to an explicit ODE system. In this work, we first introduce a new numerical method for the numerical integration of this limiting ODE system, which requires to solve a linear second order PDE at each time step. We also provide a rigorous theoretical justification for this method that applies to a general class of 2D domains. We then develop and analyze a numerical strategy based on the finite-dimensional ODE system to efficiently simulate the infinite-dimensional TDGL equations in the presence of a constant external magnetic field and for small, but finite, $\epsilon$. This method allows us to avoid resolving the $\epsilon$-scale when solving the TDGL equations, where small values of $\epsilon$ typically require very fine meshes and time steps. We provide numerical examples on a few test cases and justify the accuracy of the method with numerical investigations.

翻译：本文研究二维含时磁性Ginzburg-Landau（TDGL）方程在较小但有限的金兹堡-朗道参数倒数ε及恒定外加磁场（关于ε的量级为1）条件下的数值模拟。在此参数范围内，TDGL方程的显著特征是出现核心尺寸量级为ε的量子化涡旋。此外，在奇异极限ε↘0下，这些涡旋遵循显式常微分方程（ODE）系统演化。本研究首先提出一种用于该极限ODE系统数值积分的新方法，该方法需要在每个时间步求解线性二阶偏微分方程（PDE），并针对一般二维区域给出了严格的理论证明。随后，基于该有限维ODE系统，我们开发并分析了一种数值策略，用于在恒定外磁场及较小有限ε条件下高效模拟无限维TDGL方程。该方法避免了在求解TDGL方程时解析ε尺度的问题——通常极小的ε值需要极精细的网格与时间步长。我们通过若干测试案例提供数值算例，并借助数值研究验证了该方法的精度。