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

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

Weizhu Bao 1*, Qinglin Tang 1

1 Department of Mathematics and Center for Computational Science and Engineering, National University of Singapore, Singapore 119076.

Received 25 January 2012; Accepted (in revised version) 6 December 2012
Available online 6 March 2013


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 $\varepsilon>0$ 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 $\varepsilon$ 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.

AMS subject classifications: 35Q56, 65M06, 65M70, 82D55

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Key words: Ginzburg-Landau equation, quantized vortex, Dirichlet boundary condition, homogeneous Neumann boundary condition, reduced dynamical laws, time splitting, compact finite difference method, finite element method.

*Corresponding author.
Email: (W. Bao), (Q. Tang)

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