A New Mixed Formulation and Efficient Numerical Solution of Ginzburg--Landau Equations Under the Temporal Gauge
Bibliographic record
Abstract
In this paper, we present a new numerical approach to the time-dependent Ginzburg--Landau (GL) equations under the temporal gauge (zero electric potential gauge). The approach is based on a mixed formulation of the GL equations, which consists of two parabolic equations for the order parameter $\psi$ and the magnetic field $\sigma = \mathrm{curl} \, \mathbf{A}$, respectively, and a vector ordinary differential equation for the magnetic potential $\mathbf{A}$. A fully linearized Galerkin finite element method is presented for solving the mixed GL system. The new approach offers many advantages on both accuracy and efficiency over existing methods. In particular, the equations for $\psi$ and $\sigma$ are uniformly parabolic and, therefore, the method provides optimal-order accuracy for the two physical components $\psi$ and $\sigma$. Since in the temporal direction, a fully linearized backward Euler scheme is used for $\psi$ and $\sigma$ and a forward Euler scheme is used for $\mathbf{A}$, respectively, the system is fully decoupled and at each time step, the three variables $\psi$, $\sigma$, and $\mathbf{A}$ can be solved simultaneously. Moreover, we present numerical comparisons with two commonly used Galerkin methods for the GL equations under the temporal gauge and the Lorentz gauge, respectively. Our numerical results show that the new approach requires fewer iterations for solving the linear systems arising at each time step and the computational cost for the vector ODE seems neglectable. Several numerical examples in both two- and three-dimensional spaces are investigated.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".