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

Erwan Faou; Georg Maierhofer; Katharina Schratz

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arXiv · 2505.03271

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

Erwan Faou·Georg Maierhofer·Katharina Schratz

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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.

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Erwan Faou, Georg Maierhofer, Katharina Schratz. 2025-05-06. Fully discrete backward error analysis for the midpoint rule applied to the nonlinear Schroedinger equation. https://arxiv.org/abs/2505.03271

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