A first-order Fourier integrator for the nonlinear Schr\"odinger equation on \mathbb{T} without loss of regularity

In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schrödinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we prove that the new scheme provides the first-order accuracy in H^\gamma for any initial data belonging to H^\gamma, for any \gamma >\frac 32. That is, up to some fixed time T, there exists some constant C=C(\|u\|_{L^\infty ([0,T]; H^{\gamma })})>0, such that \begin{equation*} \|u^n-u(t_n)\|_{H^\gamma (\mathbb {T})}\le C \tau , \end{equation*} where u^n denotes the numerical solution at t_n=n\tau. Moreover, the mass of the numerical solution M(u^n) verifies \begin{equation*} \left |M(u^n)-M(u_0)\right |\le C\tau ^5. \end{equation*} In particular, our scheme does not cost any additional derivative for the first-order convergence and the numerical solution obeys the almost mass conservation law. Furthermore, if u_0\in H^1(\mathbb {T}), we rigorously prove that \begin{equation*} \|u^n-u(t_n)\|_{H^1(\mathbb {T})}\le C\tau ^{\frac 12-}, \end{equation*} where C= C(\|u_0\|_{H^1(\mathbb {T})})>0.