For the numerical solution of the cubic nonlinear Schr\"{o}dinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in $H^s(\mathbb T^2)$, where $s>0$, convergence of order $\mathcal O(\tau^{s/2}+N^{-s})$ is proved in $L^2$. Here $\tau$ denotes the time step size and $N$ the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces, the final convergence result, however, is given in $L^2$. The stated convergence behavior is illustrated by several numerical examples.

翻译：针对具有周期性边界条件的三次非线性薛定谔方程的数值求解，本文考虑了一种空间方向采用伪谱方法、时间方向采用滤波李分裂格式的数值方案。我们证明该格式即使在初始数据具有极低正则性的情况下仍能收敛。特别地，对于$H^s(\mathbb T^2)$（其中$s>0$）中的初始数据，我们在$L^2$范数下证明了$\mathcal O(\tau^{s/2}+N^{-s})$阶的收敛性。此处$\tau$表示时间步长，$N$表示所考虑的傅里叶模态数。该结果的证明在离散Bourgain空间的抽象框架下完成，但最终收敛结果是在$L^2$空间中给出的。通过多个数值算例验证了上述收敛行为。