A fully discrete and fully explicit low-regularity integrator is constructed for the one-dimensional periodic cubic nonlinear Schrödinger equation. The method can be implemented by using fast Fourier transform with \(O(N\ln N)\) operations at every time level, and is proved to have an \(L^2\)-norm error bound of \(O(\tau \sqrt{\ln (1/\tau )}+N^{-1})\) for \(H^1\) initial data, without requiring any CFL condition, where \(\tau \) and N denote the temporal stepsize and the degree of freedoms in the spatial discretisation, respectively.

中文翻译：

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一维周期三次非线性薛定谔方程的全离散低正则积分器

为一维周期三次非线性薛定谔方程构造了一个完全离散和完全显式的低正则积分器。该方法可以通过使用快速傅立叶变换在每个时间级别进行\(O(N\ln N)\)运算来实现，并证明具有\(L^2\) -范数误差界为\(O( \tau \sqrt{\ln (1/\tau )}+N^{-1})\)用于\(H^1\)初始数据，不需要任何 CFL 条件，其中\(\tau \)和N分别表示空间离散化中的时间步长和自由度。