Plane wave stability of some conservative schemes for the cubic Schr\"{o}dinger equation

TL;DR

This paper analyzes the stability of conservative numerical schemes for the cubic Schrödinger equation, revealing differences between methods and proposing an improved energy-preserving scheme with better stability.

Contribution

It provides a comparative stability analysis of Besse and Fei et al. schemes and introduces a generalized energy-preserving method with enhanced stability properties.

Findings

Besse and Fei schemes exhibit different stability behaviors.

The generalized energy-preserving scheme shows improved stability.

Analysis enhances understanding of scheme stability for the cubic Schrödinger equation.

Abstract

The plane wave stability properties of the conservative schemes of Besse and Fei et al. for the cubic Schr\"{o}dinger equation are analysed. Although the two methods possess many of the same conservation properties, we show that their stability behaviour is very different. An energy preserving generalisation of the Fei method with improved stability is presented.