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arXiv · 2404.02133

Numerical simulation of the Gross-Pitaevskii equation via vortex tracking

Thiago Carvalho Corso·Gaspard Kemlin·Christof Melcher·Benjamin Stamm
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Abstract

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.

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Thiago Carvalho Corso, Gaspard Kemlin, Christof Melcher, Benjamin Stamm. 2024-04-02. Numerical simulation of the Gross-Pitaevskii equation via vortex tracking. https://doi.org/10.1090/mcom/4071

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