Numerical integrators for continuous disordered nonlinear Schr\"odinger equation

TL;DR

This paper introduces a low-regularity integrator for the continuous disordered nonlinear Schrödinger equation, effectively handling rough potentials and achieving second-order accuracy, with verified numerical performance.

Contribution

The paper applies a low-regularity integrator to the disordered nonlinear Schrödinger equation, overcoming regularity issues and demonstrating improved accuracy over classical methods.

Findings

LRI achieves second-order accuracy in $L^2$-norm for potentials in $H^2$.

Numerical experiments confirm theoretical convergence rates.

LRI outperforms classical methods under rougher random potentials.

Abstract

In this paper, we consider the numerical solution of the continuous disordered nonlinear Schr\"odinger equation, which contains a spatial random potential. We address the finite time accuracy order reduction issue of the usual numerical integrators on this problem, which is due to the presence of the random/rough potential. By using the recently proposed low-regularity integrator (LRI) from (33, SIAM J. Numer. Anal., 2019), we show how to integrate the potential term by losing two spatial derivatives. Convergence analysis is done to show that LRI has the second order accuracy in $L^2$-norm for potentials in $H^2$. Numerical experiments are done to verify this theoretical result. More numerical results are presented to investigate the accuracy of LRI compared with classical methods under rougher random potentials from applications.