A fully discrete low-regularity integrator for the nonlinear Schr\"odinger equation

TL;DR

This paper introduces a fully discrete, explicit low-regularity integrator for the cubic nonlinear Schrödinger equation in one dimension, achieving efficient computation and provable error bounds for initial data with low regularity.

Contribution

It presents a novel explicit low-regularity integrator with proven error bounds and efficient implementation for the nonlinear Schrödinger equation.

Findings

Achieves an error bound of O(τ^{1.5γ - 0.5 - ε} + N^{-γ}) in L^2 norm.

Efficient implementation using FFT with O(N log N) complexity per step.

Numerical examples confirm the theoretical convergence rates.

Abstract

For the solution of the cubic nonlinear Schr\"odinger equation in one space dimension, we propose and analyse a fully discrete low-regularity integrator. The scheme is explicit and can easily be implemented using 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 discretisation. We prove that the new scheme provides an $\mathcal{O}(\tau^{\frac32\gamma-\frac12-\varepsilon}+N^{-\gamma})$ error bound in $L^2$ for any initial data belonging to $H^\gamma$, $\frac12<\gamma\leq 1$, where $\tau$ denotes the temporal step size. Numerical examples illustrate this convergence behavior.