Numerical approximations of the Ginzburg–Landau models for superconductivity

Du, Qiang

doi:10.1063/1.2012127

Cited by 80 publications

(71 citation statements)

References 88 publications

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“…For example, when α = 1 and β = 0, it collapses into the nonlinear heat equation (NLHE) or the Ginzburg-Landau equation (GLE) [27,28]. The GLE with a complex order parameter is well known for modeling superconductivity [10,11,14,12,19], while that with a real order parameter corresponds to the Allen-Cahn equation in phase transition [13]. When α = 0 and β = 1, the GLSE reduces to the nonlinear Schrödinger equation (NLSE) [27,31,22] for modeling, for example, superfluidity or Bose-Eistein condensation (BEC).…”

Section: Introductionmentioning

confidence: 99%

The Dynamics and Interaction of Quantized Vortices in the Ginzburg–Landau–Schrödinger Equation

Zhang¹,

Bao²,

Du³

2007

SIAM J. Appl. Math.

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Abstract. The dynamic laws of quantized vortex interactions in the Ginzburg-Landau-Schrödinger equation (GLSE) are analytically and numerically studied. A review of the reduced dynamic laws governing the motion of vortex centers in the GLSE is provided. The reduced dynamic laws are solved analytically for some special initial data. By directly simulating the GLSE with an efficient and accurate numerical method proposed recently in [Y. Zhang, W. Bao, and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation, European J. Appl. Math., to appear], we can qualitatively and quantitatively compare quantized vortex interaction patterns of the GLSE with those from the reduced dynamic laws. Some conclusive findings are obtained, and discussions on numerical and theoretical results are made to provide further understanding of vortex interactions in the GLSE. Finally, the vortex motion under an inhomogeneous potential in the GLSE is also studied.

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Section: Introductionmentioning

confidence: 99%

The Dynamics and Interaction of Quantized Vortices in the Ginzburg–Landau–Schrödinger Equation

Zhang¹,

Bao²,

Du³

2007

SIAM J. Appl. Math.

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“…Refer to [14] for a recent review of methods. Here we use the Sobolev trust-region algorithm, which we think of as an enhanced Sobolev gradient descent method, to minimize the functional.…”

Section: Ginzburg-landau Energy Functionalmentioning

confidence: 99%

Minimization of the Ginzburg–Landau energy functional by a Sobolev gradient trust-region method

Kazemi

Renka

2013

Applied Mathematics and Computation

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We describe a generalized Levenberg-Marquardt method for computing critical points of the Ginzburg-Landau energy functional which models superconductivity. The algorithm is a blend of a Newton iteration with a Sobolev gradient descent method, and is equivalent to a trust-region method in which the trustregion radius is defined by a Sobolev metric. Numerical test results demonstrate the method to be remarkably effective.

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“…Until recently, this method usually has been limited to 2D simulation [6][7][8][9] or small 3D simulation [10]. Now work has been initiated, however, to implement large 3D simulations where macroscale phenomena can be observed [11,12] taking into account the collective dynamics of many vortices.…”

Section: Introductionmentioning

confidence: 99%

Detecting vortices in superconductors: Extracting one-dimensional topological singularities from a discretized complex scalar field

et al. 2015

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In type-II superconductors, the dynamics of superconducting vortices determine their transport properties. In the Ginzburg-Landau theory, vortices correspond to topological defects in the complex order parameter. Extracting their precise positions and motion from discretized numerical simulation data is an important, but challenging task. In the past, vortices have mostly been detected by analyzing the magnitude of the complex scalar field representing the order parameter and visualized by corresponding contour plots and isosurfaces. However, these methods, primarily used for small-scale simulations, blur the fine details of the vortices, scale poorly to large-scale simulations, and do not easily enable isolating and tracking individual vortices. Here we present a method for exactly finding the vortex core lines from a complex order parameter field. With this method, vortices can be easily described at a resolution even finer than the mesh itself. The precise determination of the vortex cores allows the interplay of the vortices inside a model superconductor to be visualized in higher resolution than has previously been possible. By representing the field as the set of vortices, this method also massively reduces the data footprint of the simulations and provides the data structures for further analysis and feature tracking.

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Numerical approximations of the Ginzburg–Landau models for superconductivity

Cited by 80 publications

References 88 publications

The Dynamics and Interaction of Quantized Vortices in the Ginzburg–Landau–Schrödinger Equation

The Dynamics and Interaction of Quantized Vortices in the Ginzburg–Landau–Schrödinger Equation

Minimization of the Ginzburg–Landau energy functional by a Sobolev gradient trust-region method

Detecting vortices in superconductors: Extracting one-dimensional topological singularities from a discretized complex scalar field

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