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

Erwan Faou, Georg Maierhofer, Katharina Schratz

TL;DR

The paper addresses long‑time behavior of symplectic integrators for Hamiltonian PDEs by establishing a dimension‑independent backward error analysis for the fully discrete nonlinear Schrödinger equation using the midpoint rule. It develops a formal and rigorous construction of a modified Hamiltonian, $H_h^{(N)}$, whose flow closely approximates the midpoint update uniformly in spatial discretization parameters, under a CFL constraint. This yields almost‑global stability for small data and provides quantitative control of energy drift, extending finite‑dimensional backward error results to PDE discretizations. The work advances the understanding of long‑time numerical behavior for Hamiltonian PDEs and supports reliable simulations of NLSE dynamics across varying spatial resolutions.

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.

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

TL;DR

The paper addresses long‑time behavior of symplectic integrators for Hamiltonian PDEs by establishing a dimension‑independent backward error analysis for the fully discrete nonlinear Schrödinger equation using the midpoint rule. It develops a formal and rigorous construction of a modified Hamiltonian, , whose flow closely approximates the midpoint update uniformly in spatial discretization parameters, under a CFL constraint. This yields almost‑global stability for small data and provides quantitative control of energy drift, extending finite‑dimensional backward error results to PDE discretizations. The work advances the understanding of long‑time numerical behavior for Hamiltonian PDEs and supports reliable simulations of NLSE dynamics across varying spatial resolutions.

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.
Paper Structure (13 sections, 13 theorems, 165 equations)

This paper contains 13 sections, 13 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Problem setting and notation
  3. The discrete NLSE
  4. Midpoint rule
  5. Functional setting
  6. Hamiltonian formulation
  7. Estimating polynomial vector fields
  8. Vector fields and Hamiltonian functions
  9. The midpoint rule as a splitting method
  10. Modified energy for the midpoint rule
  11. Formal construction of the modified energy
  12. Final estimates and modified energy
  13. Application to long-time stability

Key Result

Lemma 2.1

$V_{\delta x}$ with the norm $\|\cdot\|\left.\newline\!\!\right._{{\delta x}}$ is an algebra, with a constant independent of dimension. In particular, for any $v,w\in V_{{\delta x}}$, where $C$ does not depend on $K$ and ${\delta x}$ and where $\bullet$ denotes the elementwise product of the two vectors. Occasionally we will drop the notation $\bullet$ when it is clear from context.

Theorems & Definitions (36)

  • Lemma 2.1
  • proof
  • Definition 2.2: Hamiltonian formulation
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6: Commutator of vector fields
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • ...and 26 more