A Fully Discrete Low-Regularity Integrator for the Nonlinear Schrödinger Equation

Abstract

For the solution of the one dimensional cubic nonlinear Schrödinger equation on the torus, we propose and analyze a fully discrete low-regularity integrator. The considered scheme is explicit. Its implementation relies on the fast Fourier transform with a complexity of ${\mathcal {O}}(N\log N)$ operations per time step, where N denotes the degrees of freedom in the spatial discretization. We prove that the new scheme provides an ${\mathcal {O}}(\tau ^{\frac{3}{2}\gamma -\frac{1}{2}-\varepsilon }+N^{-\gamma })$ error bound in $L^2$ for any initial data in $H^\gamma$, $\frac{1}{2}<\gamma \le 1$, where $\tau$ denotes the temporal step size. Numerical examples illustrate this convergence behavior.