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.

翻译：本文研究Gross-Pitaevskii（GP）方程的数值模拟问题，该方程的一个显著特征是会出现核心尺寸为小参数$\varepsilon$量级的量子化涡旋。在无磁场且具有适当初始条件的情况下，这些涡旋在奇异极限$\varepsilon\to0$下通过显式哈密顿动力学发生相互作用。基于这一解析框架，我们开发并分析了一种基于降阶哈密顿系统的数值策略，以高效模拟小参数$\varepsilon$有限但趋小时的无限维GP方程。该方法使我们能够避免直接求解GP方程时遇到的数值稳定性问题——在$\varepsilon$取值较小时，通常需要极精细的网格划分与时间步长。我们通过超电流误差的严格估计给出了该方法的数学论证，并辅以数值算例加以说明。