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 Schr\"odinger 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 $\gamma >\frac32$. That is, up to some fixed time $T$, there exists some constant $C=C(\|u\|_{L^\infty([0,T]; H^{\gamma})})>0$, such that $$ \|u^n-u(t_n)\|_{H^\gamma(\mathbb T)}\le C \tau, $$ where $u^n$ denotes the numerical solution at $t_n=n\tau$. Moreover, the mass of the numerical solution $M(u^n)$ verifies $$ \left|M(u^n)-M(u_0)\right|\le C\tau^5. $$ In particular, our scheme dose 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 $$ \|u^n-u(t_n)\|_{H^1(\mathbb T)}\le C\tau^{\frac12-}, $$ where $C= C(\|u_0\|_{H^1(\mathbb T)})>0$.