Nonlinear Schrödinger equations (NLS) are crucial in quantum physics. Over the past century, they have been widely applied in diverse fields, such as the propagation of laser beams, water waves, and the studies of Bose--Einstein condensates. As the equations can't be solved explicitly, numerical solutions become important as well. Therefore, over the past decades, numerical methods for NLS have been extensively studied in the literature.
In this paper, we consider the cubic NLS on a $d$-dimensional torus $\mathbb{T}^d$, i.e. with periodic boundary conditions. As one of the most popular cases of NLS, multitudes of traditional methods are developed. However, these traditional methods require a lot of regularity for the stability argument. Such restriction is not necessary for the local or global well-posedness of the solution. To be specific, $s>\frac d2+2$ is required in traditional methods, where $s$ denotes the regularity of the initial data $u_0$, i.e. $u_0\in H^s(\mathbb{T}^d)$. However, the local well-posedness of the solution only needs $s>\frac d2-1$. Moreover, traditional methods will fail when this restriction is not satisfied, e.g., simulating the Talbot effect with jump discontinuities. The numerical analysis of this low regularity case is still an open question.
To overcome such restrictions, we developed a filtered Lie--Trotter splitting method for the cubic NLS on torus. This method shows very good properties in both convergence and long-time behaviours. Moreover, Bourgain developed Bourgain techniques for the analysis of the method, and we adapted these techniques in numerical analysis by constructing discrete Bourgain spaces. These techniques allow us to handle the error analysis for the case $s>\max(\frac d2-1,0)$, which coincides with the regularity restrictions of local well-posedness of the exact solution for $d\geq2$. We also proved that the filtered Lie splitting is convergent in $L^2$, with a convergence order of $\frac s2$ in time and $s$ in space. Our numerical solutions illustrate that this result is sharp.