Numerical computations for motion of vortices governed by a hyperbolic Ginzburg-Landau system (Q697519)

This paper deals with the problem \(u_{tt}-\Delta u = (2/\delta)(1-| u| ^ 2)u\) on a Lipschitz domain of \(\mathbb R^M\) with values in \(\mathbb R^N\), where \(\delta\) is a positive constant. Numerical computations on a square (\(M=N=2\), \(\delta=0.1\)) are presented, illustrating bouncing vortices (if Dirichlet conditions are imposed), and reflection of vortices at the boundary (in the case of Neumann conditions). Some results for \(M=N=3\) are also reported. The method consists in solving a semi-discretized, implicit scheme by a variational method. Some estimates on semi-discretized problems, as \(\Delta t\to 0\), are given.