SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2695-2717, December 2023.
Abstract. 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 Nedélec 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.

中文翻译：

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时态规范下动态Ginzburg-Landau方程的能量稳定和最大界原理守恒方案

《SIAM 数值分析杂志》，第 61 卷，第 6 期，第 2695-2717 页，2023 年 12 月。
摘要。本文提出了时间规范下瞬态Ginzburg-Landau方程的解耦数值格式。对于磁势和序参数，离散方案分别采用第二类Nedélec单元和线性单元进行空间离散；以及分别用于时间离散化的线性化后向欧拉方法和一阶指数时间差分方法。证明了有序参数的最大界原理（MBP）和离散意义上的能量耗散定律。离散能量稳定性和MBP保存可以保证数值模拟的稳定性和有效性，并进一步促进自适应时间步进策略的采用，该策略在涡动力学的长时间模拟中往往发挥着重要作用，特别是当应用磁场很强。还给出了所提出方案的最佳误差估计。数值算例验证了该方案的理论结果，并演示了超导体在外部磁场中的涡旋运动。