Volume 14, Issue 3
Numerical Study of Quantized Vortex Interaction in the Ginzburg-Landau Equation on Bounded Domains

Weizhu Bao & Qinglin Tang

Commun. Comput. Phys., 14 (2013), pp. 819-850.

Published online: 2013-09

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  • Abstract

Inthis paper,we study numerically quantized vortex dynamics and their interactionin the two-dimensional (2D)Ginzburg-Landauequation (GLE)with adimensionless parameter ε>0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a reviewof the reduceddynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically. Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws, we simulate quantized vortex interaction of GLE with different ε and under different initial setups including single vortex, vortex pair, vortex dipole and vortex lattice, compare them with those obtained from the corresponding reduced dynamical laws, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction. Finally, we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.


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@Article{CiCP-14-819, author = {}, title = {Numerical Study of Quantized Vortex Interaction in the Ginzburg-Landau Equation on Bounded Domains}, journal = {Communications in Computational Physics}, year = {2013}, volume = {14}, number = {3}, pages = {819--850}, abstract = {

Inthis paper,we study numerically quantized vortex dynamics and their interactionin the two-dimensional (2D)Ginzburg-Landauequation (GLE)with adimensionless parameter ε>0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a reviewof the reduceddynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically. Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws, we simulate quantized vortex interaction of GLE with different ε and under different initial setups including single vortex, vortex pair, vortex dipole and vortex lattice, compare them with those obtained from the corresponding reduced dynamical laws, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction. Finally, we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.


}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.250112.061212a}, url = {http://global-sci.org/intro/article_detail/cicp/7183.html} }
TY - JOUR T1 - Numerical Study of Quantized Vortex Interaction in the Ginzburg-Landau Equation on Bounded Domains JO - Communications in Computational Physics VL - 3 SP - 819 EP - 850 PY - 2013 DA - 2013/09 SN - 14 DO - http://dor.org/10.4208/cicp.250112.061212a UR - https://global-sci.org/intro/article_detail/cicp/7183.html KW - AB -

Inthis paper,we study numerically quantized vortex dynamics and their interactionin the two-dimensional (2D)Ginzburg-Landauequation (GLE)with adimensionless parameter ε>0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a reviewof the reduceddynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically. Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws, we simulate quantized vortex interaction of GLE with different ε and under different initial setups including single vortex, vortex pair, vortex dipole and vortex lattice, compare them with those obtained from the corresponding reduced dynamical laws, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction. Finally, we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.


Weizhu Bao & Qinglin Tang. (2020). Numerical Study of Quantized Vortex Interaction in the Ginzburg-Landau Equation on Bounded Domains. Communications in Computational Physics. 14 (3). 819-850. doi:10.4208/cicp.250112.061212a
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