Numerical simulation of the Gross-Pitaevskii equation via vortex tracking
<|reference_start|>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 $\varepsilon$. Without a magnetic field and with suitable initial conditions, these vortices interact, in the singular limit $\varepsilon\to0$, 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, $\varepsilon$. This method allows us to avoid numerical stability issues in solving the GP equation, where small values of $\varepsilon$ 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.<|reference_end|>
@article{corso2024numerical,
    title={Numerical simulation of the Gross-Pitaevskii equation via vortex
  tracking},
    author={Thiago Carvalho Corso (IANS-NMH, Stuttgart University), Gaspard Kemlin
  (LAMFA, UPJV), Christof Melcher (AA, RWTH) and Benjamin Stamm (IANS-NMH,
  Stuttgart University)},
    journal={arXiv preprint arXiv:2404.02133},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.02133},
    primaryClass={math.NA cs.NA}
}