An energy stable and maximum bound principle preserving scheme for the dynamic Ginzburg-Landau equations under the temporal gauge

TL;DR

This paper introduces a new numerical scheme for the dynamic Ginzburg-Landau equations that guarantees energy stability and preserves the maximum bound principle, improving long-term vortex simulation accuracy.

Contribution

It develops a decoupled, energy-stable numerical scheme with MBP preservation for the Ginzburg-Landau equations, including error analysis and practical vortex simulation validation.

Findings

The scheme maintains energy dissipation and MBP in discrete form.

Numerical results confirm the scheme's stability and accuracy.

The method effectively simulates vortex dynamics under strong magnetic fields.

Abstract

This paper proposes a decoupled numerical scheme of the time-dependent Ginzburg--Landau equations under the temporal gauge. For the magnetic potential and the order parameter, the discrete scheme adopts the second type Ned${\rm \acute{e}}$lec element and the linear element for spatial discretization, respectively; and a linearized backward Euler method and the first order exponential time differencing method for time discretization, respectively. The maximum bound principle (MBP) of the order parameter and the energy dissipation law in the discrete sense are proved. The discrete energy stability and MBP-preservation can guarantee the stability and validity of the numerical simulations, and further facilitate the adoption of an adaptive time-stepping strategy, which often plays an important role in long-time simulations of vortex dynamics, especially when the applied magnetic field is strong. An optimal error estimate of the proposed scheme is also given. Numerical examples verify the theoretical results of the proposed scheme and demonstrate the vortex motions of superconductors in an external magnetic field.