Implicit Integration of the Time-Dependent Ginzburg-Landau Equations of Superconductivity

TL;DR

This work addresses efficient integration of the time-dependent Ginzburg–Landau equations for superconductivity by comparing four time-stepping schemes from fully explicit to fully implicit on a vortex-equilibration benchmark. It develops dimensionless TDGL formulations with gauge-invariant discretizations and analyzes stability, accuracy, and compute time, demonstrating that the fully implicit method can increase the allowable time step by about $80\times$ while reducing overall wall time to equilibrium. Despite higher per-step cost, the wall-clock savings are substantial and linear parallel speedups are observed on multi-processor hardware. All schemes converge to the same equilibrium vortex configuration, validating the approaches for scalable simulations of vortex dynamics in superconductors.

Abstract

This article is concerned with the integration of the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity. Four algorithms, ranging from fully explicit to fully implicit, are presented and evaluated for stability, accuracy, and compute time. The benchmark problem for the evaluation is the equilibration of a vortex configuration in a superconductor that is embedded in a thin insulator and subject to an applied magnetic field.

Figures (4)

• Figure 1: Computational cell with evaluation points for $\psi$, $A_x$, and $A_y$.
• Figure 2: Elapsed time for 50 time steps as a function of the number of processors.
• Figure 3: Equilibrium vortex configuration for the benchmark problem.
• Figure 4: Computational cell with evaluation points for $\psi$, $A_x$, and $A_y$.