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

Boyi Wang, Saurav Shenoy, Daniel Fortino, Long-Qing Chen, Wenrui Hao

TL;DR

This work develops a structure-preserving implicit–explicit scheme for a hybrid time-dependent Ginzburg–Landau model that incorporates a nonlinear BCS gap coupling, enabling stable, long-time simulations beyond the near-critical regime. By nondimensionalizing the coupled TDGL–BCS system and deriving a maximum modulus-preserving, energy-stable IMEX method, the authors ensure physical consistency under gauge constraints and external magnetic fields. Formal asymptotics connect the model to the classical TDGL equation as $T\to T_c$, and numerical experiments in 2D and 3D demonstrate accurate vortex formation, alignment, and superconductivity suppression under increasing fields, including effects of inhomogeneity. The proposed framework offers a robust tool for exploring vortex dynamics in superconductors across a broad temperature range with reliable stability guarantees.

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.

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

TL;DR

This work develops a structure-preserving implicit–explicit scheme for a hybrid time-dependent Ginzburg–Landau model that incorporates a nonlinear BCS gap coupling, enabling stable, long-time simulations beyond the near-critical regime. By nondimensionalizing the coupled TDGL–BCS system and deriving a maximum modulus-preserving, energy-stable IMEX method, the authors ensure physical consistency under gauge constraints and external magnetic fields. Formal asymptotics connect the model to the classical TDGL equation as , and numerical experiments in 2D and 3D demonstrate accurate vortex formation, alignment, and superconductivity suppression under increasing fields, including effects of inhomogeneity. The proposed framework offers a robust tool for exploring vortex dynamics in superconductors across a broad temperature range with reliable stability guarantees.

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.
Paper Structure (7 sections, 4 theorems, 42 equations, 6 figures, 2 tables)

This paper contains 7 sections, 4 theorems, 42 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model problem
  3. Non-dimensionalization of the Model
  4. Asymptotic behavior when $T\rightarrow T_c$
  5. Energy stable semi-discrete scheme
  6. Numerical results
  7. Discussion and Conclusion

Key Result

Proposition 2.1

Consider the hybrid TDGL system eq-m--iv-1 with the nonlinear term $f(|\psi|^2)$ defined in pot. Let $\varepsilon>0$ be the dimensionless parameter associated with the scaling $\beta = \beta_0/(1-\varepsilon^2)$ and assume the expansions Then, as $\varepsilon\to 0$, the leading-order approximation of the system formally reduces to the classical TDGL-type system where $\tau=\varepsilon^2 t$, $\ha

Figures (6)

  • Figure 1: Snapshots at $0.96\,T_c$ ($T_c$: critical temperature), initialized with a small amplitude ($|\psi(0)| \approx 0.1$). Rows 1 and 2 show the superconducting order parameter magnitude $|\psi|$ for $H=0.15$ (row 1) and $H=0.3$ (row 2), at times $t=20, 40, 60, 80, 100$. Rows 3 and 4 show the corresponding magnetic field distribution $\mathrm{curl}\,A$ for $H=0.15$ (row 3) and $H=0.3$ (row 4). Comparison of rows 1–2 indicates that the decay of $|\psi|$ is faster and reaches smaller values at $H=0.3$. Arrows indicate the direction of the vector potential $A$. The two blocks (upper and lower panels) correspond to two independent datasets (fig1 and figA1)
  • Figure 2: Time evolution of the spatially averaged order parameter corresponding to Figures \ref{['fig.3']} and \ref{['fig.4']}. In this example, superconductivity is suppressed more strongly by the larger magnetic field, leading to smaller values of the order parameter modulus.
  • Figure 3: Time evolution of vortex structures of $|\psi|$ and $curl A$ in $(-\pi,\pi)$ with GL constant $\kappa = 2$ under an external magnetic field along the (0,0,1) direction (t=20, 40, 60, 80, 100, and the arrows indicate the direction of the vector potential A). The stabilization parameter is $S = 4$. The singular and even rows correspond to the cases with and without the applied inhomogeneity potential, respectively
  • Figure 4: Time evolution of vortex structures of $|\psi|$ and $curl A$ in $(-2\pi,2\pi)$ with GL constant $\kappa = 2$ under an external magnetic field along the (0,0,1) direction (t=20, 40, 60, 80, 100, and the arrows indicate the direction of the vector potential A). he stabilization parameter is $S = 2$. The singular and even rows correspond to the cases with and without the applied inhomogeneity potential, respectively.
  • Figure 5: Time evolution of vortex structures in a 3D cubic domain under different applied magnetic fields. Top row: $\delta(x)=0$, $H=(0,\,0,\,0.5)$; middle row: $\delta(x)=0$, $H=(0,\,0.5\sin(\pi/36),\,0.5\cos(\pi/36))$; bottom row: $\delta(x)\neq0$, $H=(0,\,0.5\sin(\pi/36),\,0.5\cos(\pi/36))$. Contours are taken at the average of the maximum and minimum values. A comparison between rows 2 and 3 shows that inhomogeneity causes vortices to deviate from the otherwise aligned structures observed in the homogeneous case.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Proposition 2.1: Formal asymptotic reduction as $T \to T_c$
  • proof : Formal derivation
  • Theorem 2.1: Convergence rate as $T \to T_c$
  • proof
  • Theorem 3.1: Maximum modulus bound
  • proof
  • Theorem 3.2: Energy inequality
  • proof
  • Remark 3.1
  • Remark 4.1