Fully discrete backward error analysis for the midpoint rule applied to the nonlinear Schroedinger equation

TL;DR

This paper develops a dimension-independent backward error analysis for the midpoint rule applied to nonlinear Schrödinger equations, demonstrating long-time stability and the existence of a modified energy regardless of spatial discretization.

Contribution

It provides the first backward error analysis for the midpoint rule on Hamiltonian PDEs, extending understanding from finite-dimensional systems to PDEs.

Findings

Existence of a modified energy for the midpoint rule on nonlinear Schrödinger equations.

Long-time stability of the numerical flow is established.

Analysis is independent of spatial discretization level.

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.