Fully discrete backward error analysis for the midpoint rule applied to the nonlinear Schroedinger equation
Analysis of PDEs
2025-05-07 v1 Numerical Analysis
Numerical Analysis
Abstract
The use of symplectic numerical schemes on Hamiltonian systems is widely known to lead to favorable long-time behaviour. While this phenomenon is thoroughly understood in the context of finite-dimensional Hamiltonian systems, much less is known in the context of Hamiltonian PDEs. In this work we provide the first dimension-independent backward error analysis for a Runge-Kutta-type method, the midpoint rule, which shows the existence of a modified energy for this method when applied to nonlinear Schroedinger equations regardless of the level of spatial discretisation. We use this to establish long-time stability of the numerical flow for the midpoint rule.
Cite
@article{arxiv.2505.03271,
title = {Fully discrete backward error analysis for the midpoint rule applied to the nonlinear Schroedinger equation},
author = {Erwan Faou and Georg Maierhofer and Katharina Schratz},
journal= {arXiv preprint arXiv:2505.03271},
year = {2025}
}