This paper proposes a decoupled numerical scheme of the time-dependent Ginzburg--Landau equations under the temporal gauge. For the magnetic potential and the order parameter, the discrete scheme adopts the second type Ned${\rm \acute{e}}$lec element and the linear element for spatial discretization, respectively; and a linearized backward Euler method and the first order exponential time differencing method for time discretization, respectively. The maximum bound principle (MBP) of the order parameter and the energy dissipation law in the discrete sense are proved. The discrete energy stability and MBP-preservation can guarantee the stability and validity of the numerical simulations, and further facilitate the adoption of an adaptive time-stepping strategy, which often plays an important role in long-time simulations of vortex dynamics, especially when the applied magnetic field is strong. An optimal error estimate of the proposed scheme is also given. Numerical examples verify the theoretical results of the proposed scheme and demonstrate the vortex motions of superconductors in an external magnetic field.

翻译：本文提出了一种在时间规范下时间依赖型Ginzburg-Landau方程的解耦数值格式。对于磁势和序参量，离散格式分别采用第二类Ned${\rm \acute{e}}$lec元和线性元进行空间离散；时间离散分别采用线性化向后欧拉法和一阶指数时间差分法。证明了序参量的最大界原则和离散意义上的能量耗散律。离散能量稳定性与最大界原则保持性可确保数值模拟的稳定性和有效性，并进一步促进自适应时间步长策略的采用，该策略在涡旋动力学长时间模拟中（尤其当外加磁场较强时）常发挥重要作用。同时给出了所提格式的最优误差估计。数值算例验证了所提格式的理论结果，并展示了超导体在外加磁场中的涡旋运动。