Numerical simulation of the Gross-Pitaevskii equation via vortex tracking

This paper deals with the numerical simulation of the Gross-Pitaevskii (GP) equation, for which a well-known feature is the appearance of quantized vortices with core size of the order of a small parameter ε. Without a magnetic field and with suitable initial conditions, these vortices interact, in the singular limit ε→0, through an explicit Hamiltonian dynamics. Using this analytical framework, we develop and analyze a numerical strategy based on the reduced-order Hamiltonian system to efficiently simulate the infinite-dimensional GP equation for small, but finite, ε. This method allows us to avoid numerical stability issues in solving the GP equation, where small values of ε typically require very fine meshes and time steps. We also provide a mathematical justification of our method in terms of rigorous error estimates of the error in the supercurrent, together with numerical illustrations.