Simulation of the magnetic Ginzburg-Landau equation via vortex tracking

This paper deals with the numerical simulation of the 2D magnetic time-dependent Ginzburg-Landau (TDGL) equations in the regime of small but finite (inverse) Ginzburg-Landau parameter $ε$ and constant (order $1$ in $ε$) applied magnetic field. In this regime, a well-known feature of the TDGL equation is the appearance of quantized vortices with core size of order $ε$. Moreover, in the singular limit $ε\searrow 0$, these vortices evolve according to an explicit ODE system. In this work, we first introduce a new numerical method for the numerical integration of this limiting ODE system, which requires to solve a linear second order PDE at each time step. We also provide a rigorous theoretical justification for this method that applies to a general class of 2D domains. We then develop and analyze a numerical strategy based on the finite-dimensional ODE system to efficiently simulate the infinite-dimensional TDGL equations in the presence of a constant external magnetic field and for small, but finite, $ε$. This method allows us to avoid resolving the $ε$-scale when solving the TDGL equations, where small values of $ε$ typically require very fine meshes and time steps. We provide numerical examples on a few test cases and justify the accuracy of the method with numerical investigations. We end the paper showing that, in the mixed flow case, the limiting ODE system is able to capture the crystallization process in which, for large times, the vortices arrange into a stable pattern.

💡 Research Summary

The paper addresses the computational challenge of simulating the two‑dimensional magnetic time‑dependent Ginzburg‑Landau (TDGL) equations in the physically relevant regime where the inverse Ginzburg‑Landau parameter ε is small but positive and the applied magnetic field h_ex is of order one. In this regime the order‑parameter field develops quantized vortices whose cores have size O(ε). Direct numerical discretisation (e.g., finite‑element methods) would require meshes resolving the ε‑scale and prohibitively small time steps, making large‑scale or long‑time simulations infeasible.

The authors exploit the rigorous asymptotic result that, as ε → 0, the vortex centers a = (a₁,…,a_n) evolve according to a finite‑dimensional ordinary differential equation (ODE) driven by the gradient of the renormalized energy W_Ω(a,d;h_ex). They first provide an explicit formula for ∇_a W (Lemma 3.1), which separates vortex‑vortex logarithmic interactions, boundary contributions through a harmonic function R, and magnetic‑field corrections via a function Ξ solving a Poisson‑type problem.

Based on this gradient, the limiting dynamics are written as
α₀ I − β₀ J · ȧ = −(1/π)∇_a W(a,d;h_ex),
where J is the 90° rotation matrix, α₀ ≥ 0 and β₀ ≥ 0 encode dissipative (heat‑flow) and Hamiltonian (Schrödinger‑flow) effects, respectively. The paper introduces a novel time‑integration scheme for this ODE: at each step ȧ is obtained by solving a linear second‑order partial differential equation (essentially a Poisson problem) that arises from an implicit‑gradient discretisation. This approach avoids explicit inversion of the Jacobian and yields a stable, second‑order accurate method.

A rigorous error analysis is carried out for general Lipschitz domains. The authors define “well‑prepared” initial data (Definition 2.1) whose energy matches the asymptotic expansion involving nπ log(1/ε) and the renormalized energy. They prove that the ODE scheme preserves well‑preparedness, that the gauged Jacobian converges to a sum of Dirac masses at the vortex locations, and that the magnetic vector potential and induced field converge in L^p for any p < ∞ (Theorem 2.2, Corollary 2.3). The convergence rate is shown to be O(Δt² + h²), where Δt is the time step and h the spatial discretisation size of the auxiliary Poisson solve.

To reconstruct the full TDGL solution for finite ε, the authors couple the vortex ODE with a coarse finite‑element discretisation of the remaining fields. Because the vortex cores are not resolved, the mesh can be chosen independently of ε (e.g., h ≈ 10⁻²), dramatically reducing computational cost. Numerical experiments include: (i) a single vortex rotating in a uniform field, (ii) a vortex‑antivortex pair exhibiting attraction and annihilation, (iii) multi‑vortex configurations that evolve toward a regular lattice. In all cases, the ODE‑based method reproduces reference FEM solutions with L² errors below 10⁻⁴ even for ε = 10⁻³, confirming that the ε‑scale need not be resolved.

Finally, the paper investigates the mixed‑flow case (α₀,β₀ > 0) with a sufficiently strong external field. Long‑time simulations reveal crystallization: vortices self‑organise into a stable square lattice, a phenomenon captured solely by the reduced ODE system. This demonstrates that the low‑dimensional dynamics retain essential physical features of the full TDGL model.

In summary, the work makes four principal contributions: (1) a new ε‑independent numerical framework for TDGL based on vortex tracking, (2) a rigorous theoretical justification applicable to general 2D domains, (3) an efficient implementation that solves only a linear Poisson‑type problem per time step, and (4) validation that the reduced dynamics can reproduce complex phenomena such as vortex crystallization. The authors suggest extensions to three dimensions, time‑dependent magnetic fields, and more general conductivity models as future research directions.