Low regularity full error estimates for the cubic nonlinear Schr\"odinger equation

TL;DR

This paper proves low regularity convergence of a spectral and splitting scheme for the cubic nonlinear Schrödinger equation, demonstrating effective numerical solutions even with very rough initial data.

Contribution

It establishes convergence rates for a pseudospectral and filtered Lie splitting method for low-regularity initial data in the cubic nonlinear Schrödinger equation.

Findings

Convergence order of ois^{s/2}+N^{-s} in L^2 for data in H^s.

Method converges even for initial data with very low regularity.

Numerical examples confirm theoretical convergence rates.

Abstract

For the numerical solution of the cubic nonlinear Schr\"{o}dinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in $H^s(\mathbb T^2)$, where $s>0$, convergence of order $\mathcal O(\tau^{s/2}+N^{-s})$ is proved in $L^2$. Here $\tau$ denotes the time step size and $N$ the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces, the final convergence result, however, is given in $L^2$. The stated convergence behavior is illustrated by several numerical examples.