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Numerical Study of Quantized Vortex Interaction in the Ginzburg-Landau Equation on Bounded Domains

Published online by Cambridge University Press:  03 June 2015

Weizhu Bao  and
Qinglin Tang
Weizhu Bao*
Affiliation:
Department of Mathematics and Center for Computational Science and Engineering, National University of Singapore, Singapore 119076
Qinglin Tang*
Affiliation:
Department of Mathematics and Center for Computational Science and Engineering, National University of Singapore, Singapore 119076
*
Corresponding author.Email:bao@math.nus.edu.sg
Email:tqltql2010@gmail.com
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Abstract

In this paper, we study numerically quantized vortex dynamics and their interaction in the two-dimensional (2D) Ginzburg-Landau equation (GLE) with a dimensionless parameter ε > 0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a review of the reduced dynamical 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.

Keywords

35Q5665M0665M7082D55Ginzburg-Landau equationquantized vortexDirichlet boundary conditionhomogeneous Neumann boundary conditionreduced dynamical lawstime splittingcompact finite difference methodfinite element method

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Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 3 , September 2013 , pp. 819 - 850
Copyright
Copyright © Global Science Press Limited 2013

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