---
_id: '19945'
abstract:
- lang: eng
  text: Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations,
    …) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian
    systems on manifolds equipped with Poisson brackets. The Poisson structure is
    connected to conservation properties and other geometric features of solutions
    to the PDE and, therefore, of great interest for numerical integration. For the
    example of Burgers' equations and related PDEs we use Clebsch variables to lift
    the original system to a collective Hamiltonian system on a symplectic manifold
    whose structure is related to the original Lie-Poisson structure. On the collective
    Hamiltonian system a symplectic integrator can be applied. Our numerical examples
    show excellent conservation properties and indicate that the disadvantage of an
    increased phase-space dimension can be outweighed by the advantage of symplectic
    integration.
article_type: original
author:
- first_name: Robert I
  full_name: McLachlan, Robert I
  last_name: McLachlan
- first_name: Christian
  full_name: Offen, Christian
  id: '85279'
  last_name: Offen
  orcid: https://orcid.org/0000-0002-5940-8057
- first_name: Benjamin K
  full_name: Tapley, Benjamin K
  last_name: Tapley
citation:
  ama: McLachlan RI, Offen C, Tapley BK. Symplectic integration of PDEs using Clebsch
    variables. <i>Journal of Computational Dynamics</i>. 2019;6(1):111-130. doi:<a
    href="https://doi.org/10.3934/jcd.2019005">10.3934/jcd.2019005</a>
  apa: McLachlan, R. I., Offen, C., &#38; Tapley, B. K. (2019). Symplectic integration
    of PDEs using Clebsch variables. <i>Journal of Computational Dynamics</i>, <i>6</i>(1),
    111–130. <a href="https://doi.org/10.3934/jcd.2019005">https://doi.org/10.3934/jcd.2019005</a>
  bibtex: '@article{McLachlan_Offen_Tapley_2019, title={Symplectic integration of
    PDEs using Clebsch variables}, volume={6}, DOI={<a href="https://doi.org/10.3934/jcd.2019005">10.3934/jcd.2019005</a>},
    number={1}, journal={Journal of Computational Dynamics}, publisher={American Institute
    of Mathematical Sciences (AIMS)}, author={McLachlan, Robert I and Offen, Christian
    and Tapley, Benjamin K}, year={2019}, pages={111–130} }'
  chicago: 'McLachlan, Robert I, Christian Offen, and Benjamin K Tapley. “Symplectic
    Integration of PDEs Using Clebsch Variables.” <i>Journal of Computational Dynamics</i>
    6, no. 1 (2019): 111–30. <a href="https://doi.org/10.3934/jcd.2019005">https://doi.org/10.3934/jcd.2019005</a>.'
  ieee: R. I. McLachlan, C. Offen, and B. K. Tapley, “Symplectic integration of PDEs
    using Clebsch variables,” <i>Journal of Computational Dynamics</i>, vol. 6, no.
    1, pp. 111–130, 2019.
  mla: McLachlan, Robert I., et al. “Symplectic Integration of PDEs Using Clebsch
    Variables.” <i>Journal of Computational Dynamics</i>, vol. 6, no. 1, American
    Institute of Mathematical Sciences (AIMS), 2019, pp. 111–30, doi:<a href="https://doi.org/10.3934/jcd.2019005">10.3934/jcd.2019005</a>.
  short: R.I. McLachlan, C. Offen, B.K. Tapley, Journal of Computational Dynamics
    6 (2019) 111–130.
date_created: 2020-10-06T16:44:07Z
date_updated: 2022-01-06T06:54:15Z
department:
- _id: '636'
doi: 10.3934/jcd.2019005
extern: '1'
intvolume: '         6'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: http://www.aimsciences.org/article/doi/10.3934/jcd.2019005
oa: '1'
page: 111-130
publication: Journal of Computational Dynamics
publication_identifier:
  issn:
  - 2158-2505
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Symplectic integration of PDEs using Clebsch variables
type: journal_article
user_id: '85279'
volume: 6
year: '2019'
...