A new second-order low-regularity integrator for the cubic nonlinear Schrödinger equation

This article is concerned with the question of whether it is possible to construct a time discretization for the one-dimensional cubic nonlinear Schrödinger equation with second-order convergence for initial data with regularity strictly below $H^2$. We address this question with a positive answer by constructing a new second-order low-regularity integrator for the one-dimensional cubic nonlinear Schrödinger equation. The proposed method can have second-order convergence in $L^2$ for initial data in $H^{\frac 53}$, and first-order convergence up to a logarithmic factor for initial data in $H^1$. This significantly relaxes the regularity requirement for second-order approximations to the cubic nonlinear Schrödinger equation, while retaining the by far best convergence order for initial data in $H^1$. Numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed method in approximating nonsmooth solutions of the nonlinear Schrödinger equation. The numerical results show that, among the many time discretizations, the proposed method is the only one that has second-order convergence in $L^2$ for initial data strictly below $H^2$.