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.

翻译：针对含低正则势的Gross-Pitaevskii方程(GPE)空间离散问题，本文提出并分析了一种扩展傅里叶伪谱(eFP)方法。该方法通过在其离散傅里叶变换的扩展窗口中处理势函数，即使在势函数正则性较低的情况下，仍能保持关于精确解正则性的最优收敛阶，且计算成本与标准傅里叶伪谱方法相当，兼具高效性与精确性。此外，与傅里叶谱/伪谱方法类似，eFP方法可便捷地与多种主流时间积分器耦合，包括有限差分法、时间分裂法和指数型积分器。数值实验结果验证了最优误差估计的准确性及锐利性，并展示了该方法在实际计算中的高效性。