Implicit integration of the TDGL equations of superconductivity

TL;DR

The paper addresses efficient, stable integration of the time-dependent Ginzburg–Landau equations for superconductivity and vortex equilibration. It develops four time-integration schemes—from fully explicit to fully implicit—for a two-dimensional, gauge-invariant discretization and evaluates them on a vortex benchmark with $N_v=116$ under a uniform field. The key finding is that the fully implicit method permits much larger time steps, reducing wall-clock time despite higher per-step cost, and that multi-timestepping yields further speedups. The work demonstrates scalable parallel performance and provides practical guidance for simulating vortex dynamics in superconductors, with implications for large-scale 3D simulations.

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 (5)

• 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: Performance of Algorithm IV in a multiprocessing environment.
• Figure 5: Effect of the updating frequency on the evolution of the energy functional.