Higher-order splitting algorithms for solving the nonlinear Schr\"odinger equation and their instabilities

TL;DR

This paper investigates higher-order splitting algorithms for the nonlinear Schrödinger equation, revealing inherent numerical instabilities caused by high wave number noise growth, which are unavoidable in continuum models but can be mitigated in discrete systems.

Contribution

The study provides a detailed error analysis showing the origin of instabilities in splitting algorithms and establishes conditions under which these instabilities can be avoided in discrete implementations.

Findings

Fourth-order convergence verified numerically.

Instability due to exponential growth of high wave number noise.

Instability unavoidable in continuum models; mitigable in discrete models under certain conditions.

Abstract

Since the kinetic and the potential energy term of the real time nonlinear Schr\"odinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence of some well known algorithms by solving the Gross-Pitaevskii equation numerically. All such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved. A detail error analysis reveals that the noise, or elementary excitations of the nonlinear Schr\"odinger, obeys the Bogoliubov spectrum and the instability is due to the exponential growth of high wave number noises caused by the splitting process. For a continuum wave function, this instability is unavoidable no matter how small the time step. For a discrete wave function, the instability can be avoided only for $\dt k_{max}^2{<\atop\sim}2 \pi$, where $k_{max}=\pi/\Delta x$.