A Structure-Preserving Scheme for the Time-Dependent Ginzburg-Landau Model with BCS Gap Coupling

TL;DR

This paper introduces a novel structure-preserving numerical scheme for a coupled model of superconductivity that remains stable and accurate across different temperature regimes, enabling detailed simulations of vortex dynamics.

Contribution

The paper develops a maximum bound preserving, energy-stable IMEX scheme for the coupled TDGL and BCS gap equations, extending the model's applicability beyond the critical temperature.

Findings

Successfully simulates vortex formation and alignment

Captures suppression of superconductivity under magnetic fields

Ensures long-time stability and physical consistency

Abstract

We propose a structure-preserving scheme for a hybrid model that couples the time-dependent Ginzburg-Landau (TDGL) equation of superconducting vortex dynamics and the nonlinear Bardeen-Cooper-Schrieffer (BCS) gap equation. This formulation is consistent with the classical TDGL equation in the near-critical temperature, while extending the applicability of the existing TDGL model to regimes beyond the critical temperature. The resulting system poses significant computational challenges due to its nonlinear and coupled structure. To achieve stable and reliable simulations of the vortex dynamics and accompanying morphological transitions, we develop a maximum bound preserving, energy-stable implicit-explicit (IMEX) scheme. The structure-preserving properties of the scheme are rigorously established, ensuring long-time stability and physical consistency. Through two- and three-dimensional simulations, the hybrid model successfully captures the temporal and spatial formation and alignment of vortices and the suppression of superconductivity under increasing magnetic fields, demonstrating both the accuracy and robustness of the proposed computational approach.