Numerical Study of Quantized Vortex Interaction in the Ginzburg-Landau Equation on Bounded Domains (Communications in Computational Physics, 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 ε>0 on bounded domains under either Dirichlet or homogeneousNeumann boundary condition. We begin with a review of the reduced dynamical lawsfor time evolution of quantized vortex centers in GLE and show how to solve thesenonlinear ordinary differential equations numerically. Then we present efficient andaccurate numerical methods for discretizing the GLE on either a rectangular or a diskdomain under either Dirichlet or homogeneous Neumann boundary condition. Basedon these efficient and accurate numerical methods for GLE and the reduced dynamical laws, we simulate quantized vortex interaction of GLE with different ε and underdifferent initial setups including single vortex, vortex pair, vortex dipole and vortexlattice, compare them with those obtained from the corresponding reduced dynamical laws, and identify the cases where the reduced dynamical laws agree qualitativelyand/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 forquantized vortex lattices under the GLE dynamics on bounded domains.