---
_id: '29236'
abstract:
- lang: eng
  text: The numerical solution of an ordinary differential equation can be interpreted
    as the exact solution of a nearby modified equation. Investigating the behaviour
    of numerical solutions by analysing the modified equation is known as backward
    error analysis. If the original and modified equation share structural properties,
    then the exact and approximate solution share geometric features such as the existence
    of conserved quantities. Conjugate symplectic methods preserve a modified symplectic
    form and a modified Hamiltonian when applied to a Hamiltonian system. We show
    how a blended version of variational and symplectic techniques can be used to
    compute modified symplectic and Hamiltonian structures. In contrast to other approaches,
    our backward error analysis method does not rely on an ansatz but computes the
    structures systematically, provided that a variational formulation of the method
    is known. The technique is illustrated on the example of symmetric linear multistep
    methods with matrix coefficients.
article_type: original
author:
- first_name: Robert
  full_name: McLachlan, Robert
  last_name: McLachlan
- first_name: Christian
  full_name: Offen, Christian
  id: '85279'
  last_name: Offen
  orcid: 0000-0002-5940-8057
citation:
  ama: McLachlan R, Offen C. Backward error analysis for conjugate symplectic methods.
    <i>Journal of Geometric Mechanics</i>. 2023;15(1):98-115. doi:<a href="https://doi.org/10.3934/jgm.2023005">10.3934/jgm.2023005</a>
  apa: McLachlan, R., &#38; Offen, C. (2023). Backward error analysis for conjugate
    symplectic methods. <i>Journal of Geometric Mechanics</i>, <i>15</i>(1), 98–115.
    <a href="https://doi.org/10.3934/jgm.2023005">https://doi.org/10.3934/jgm.2023005</a>
  bibtex: '@article{McLachlan_Offen_2023, title={Backward error analysis for conjugate
    symplectic methods}, volume={15}, DOI={<a href="https://doi.org/10.3934/jgm.2023005">10.3934/jgm.2023005</a>},
    number={1}, journal={Journal of Geometric Mechanics}, publisher={AIMS Press},
    author={McLachlan, Robert and Offen, Christian}, year={2023}, pages={98–115} }'
  chicago: 'McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate
    Symplectic Methods.” <i>Journal of Geometric Mechanics</i> 15, no. 1 (2023): 98–115.
    <a href="https://doi.org/10.3934/jgm.2023005">https://doi.org/10.3934/jgm.2023005</a>.'
  ieee: 'R. McLachlan and C. Offen, “Backward error analysis for conjugate symplectic
    methods,” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, pp. 98–115, 2023,
    doi: <a href="https://doi.org/10.3934/jgm.2023005">10.3934/jgm.2023005</a>.'
  mla: McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate
    Symplectic Methods.” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, AIMS
    Press, 2023, pp. 98–115, doi:<a href="https://doi.org/10.3934/jgm.2023005">10.3934/jgm.2023005</a>.
  short: R. McLachlan, C. Offen, Journal of Geometric Mechanics 15 (2023) 98–115.
date_created: 2022-01-11T12:48:39Z
date_updated: 2023-08-10T08:40:30Z
ddc:
- '510'
department:
- _id: '636'
doi: 10.3934/jgm.2023005
external_id:
  arxiv:
  - '2201.03911'
file:
- access_level: open_access
  content_type: application/pdf
  creator: coffen
  date_created: 2022-08-12T16:48:59Z
  date_updated: 2022-08-12T16:48:59Z
  description: The numerical solution of an ordinary differential equation can be
    interpreted as the exact solution of a nearby modified equation. Investigating
    the behaviour of numerical solutions by analysing the modified equation is known
    as backward error analysis. If the original and modified equation share structural
    properties, then the exact and approximate solution share geometric features such
    as the existence of conserved quantities. Conjugate symplectic methods preserve
    a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian
    system. We show how a blended version of variational and symplectic techniques
    can be used to compute modified symplectic and Hamiltonian structures. In contrast
    to other approaches, our backward error analysis method does not rely on an ansatz
    but computes the structures systematically, provided that a variational formulation
    of the method is known. The technique is illustrated on the example of symmetric
    linear multistep methods with matrix coefficients.
  file_id: '32801'
  file_name: BEA_MultiStep_Matrix.pdf
  file_size: 827030
  relation: main_file
  title: Backward error analysis for conjugate symplectic methods
file_date_updated: 2022-08-12T16:48:59Z
has_accepted_license: '1'
intvolume: '        15'
issue: '1'
keyword:
- variational integrators
- backward error analysis
- Euler--Lagrange equations
- multistep methods
- conjugate symplectic methods
language:
- iso: eng
oa: '1'
page: 98-115
publication: Journal of Geometric Mechanics
publication_status: published
publisher: AIMS Press
quality_controlled: '1'
related_material:
  link:
  - relation: software
    url: https://github.com/Christian-Offen/BEAConjugateSymplectic
status: public
title: Backward error analysis for conjugate symplectic methods
type: journal_article
user_id: '85279'
volume: 15
year: '2023'
...