Numerical simulation of the Gross-Pitaevskii equation via vortex tracking
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915563755798528 |
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| author | Corso, Thiago Carvalho Kemlin, Gaspard Melcher, Christof Stamm, Benjamin |
| author_facet | Corso, Thiago Carvalho Kemlin, Gaspard Melcher, Christof Stamm, Benjamin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02133 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical simulation of the Gross-Pitaevskii equation via vortex tracking Corso, Thiago Carvalho Kemlin, Gaspard Melcher, Christof Stamm, Benjamin Numerical Analysis 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. |
| title | Numerical simulation of the Gross-Pitaevskii equation via vortex tracking |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2404.02133 |