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

Abstract:

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 $C= C(\|u_0\|_{H^1(\mathbb {T})})>0$.