针对具有低正则性势的Gross-Pitaevskii方程的一种扩展傅里叶伪谱方法
An extended Fourier pseudospectral method for the Gross-Pitaevskii equation with low regularity potential
AI总结：
本文提出一种扩展傅里叶伪谱方法，用于求解具有低正则性势的Gross-Pitaevskii方程，在保持最优收敛率和计算效率的同时能灵活耦合多种时间积分器，并通过数值实验验证了其准确性与高效性。
摘要：
我们提出并分析了一种扩展傅里叶伪谱（eFP）方法，用于对具有低正则性势的Gross-Pitaevskii方程（GPE）进行空间离散化，其处理方式是在扩展窗口中对势进行离散傅里叶变换。所提出的eFP方法即使势具有低正则性，也能保持相对于精确解正则性的最优收敛率，且计算成本与标准傅里叶伪谱方法相似，因此兼具高效性与准确性。此外，与傅里叶谱/伪谱方法类似，eFP方法能轻松耦合多种常用时间积分器，包括有限差分方法、时间分裂方法和指数型积分器。我们给出了数值结果以验证最优误差估计，证明其误差估计是锐利的，并展示了其在实际计算中的高效性。
英文摘要：
We propose and analyze an extended Fourier pseudospectral (eFP) method for the spatial discretization of the Gross-Pitaevskii equation (GPE) with low regularity potential by treating the potential in an extended window for its discrete Fourier transform. The proposed eFP method maintains optimal convergence rates with respect to the regularity of the exact solution even if the potential is of low regularity and enjoys similar computational cost as the standard Fourier pseudospectral method, and thus it is both efficient and accurate. Furthermore, similar to the Fourier spectral/pseudospectral methods, the eFP method can be easily coupled with different popular temporal integrators including finite difference methods, time-splitting methods and exponential-type integrators. Numerical results are presented to validate our optimal error estimates and to demonstrate that they are sharp as well as to show its efficiency in practical computations.
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