Simulation of the magnetic Ginzburg-Landau equation via vortex tracking

This paper deals with the numerical simulation of the 2D magnetic time-dependent Ginzburg-Landau (TDGL) equations in the regime of small but finite (inverse) Ginzburg-Landau parameter $\epsilon$ and constant (order $1$ in $\epsilon$) applied magnetic field. In this regime, a well-known feature of the TDGL equation is the appearance of quantized vortices with core size of order $\epsilon$. Moreover, in the singular limit $\epsilon \searrow 0$, these vortices evolve according to an explicit ODE system. In this work, we first introduce a new numerical method for the numerical integration of this limiting ODE system, which requires to solve a linear second order PDE at each time step. We also provide a rigorous theoretical justification for this method that applies to a general class of 2D domains. We then develop and analyze a numerical strategy based on the finite-dimensional ODE system to efficiently simulate the infinite-dimensional TDGL equations in the presence of a constant external magnetic field and for small, but finite, $\epsilon$. This method allows us to avoid resolving the $\epsilon$-scale when solving the TDGL equations, where small values of $\epsilon$ typically require very fine meshes and time steps. We provide numerical examples on a few test cases and justify the accuracy of the method with numerical investigations.