Communications in Mathematical Sciences

Volume 15 (2017)

Number 2

Numerical analysis and simulation for a generalized planar Ginzburg–Landau equation in a circular geometry

Pages: 329 – 357

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n2.a3

Authors

Sean Colbert-Kelly (Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, Maryland, U.S.A.)

Geoffrey B. McFadden (Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, Maryland, U.S.A.)

Daniel Phillips (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Jie Shen (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Abstract

In this paper, a numerical scheme for a generalized planar Ginzburg–Landau energy in a circular geometry is studied. A spectral-Galerkin method is utilized, and a stability analysis and an error estimate for the scheme are presented. It is shown that the scheme is unconditionally stable. We present numerical simulation results that have been obtained by using the scheme with various sets of boundary data, including those the form $u(\theta)=\exp(id\theta)$, where the integer $d$ denotes the topological degree of the solution. These numerical results are in good agreement with the experimental and analytical results. Results include the computation of bifurcations from pure bend or splay patterns to spiral patterns for $d=1$, energy decay curves for $d=1$, spectral accuracy plots for $d=2$ and computations of metastable or unstable higher-energy solutions as well as the lowest energy ground state solutions for values of $d$ ranging from two to five.

Keywords

Ginzburg–Landau, vortices, Euler–Lagrange equations, spectral-Galerkin, polar coordinates

2010 Mathematics Subject Classification

35J57, 65F05, 65N22, 65N35

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Published 21 February 2017