Comptes Rendus
no. 2
A new construction for invariant numerical schemes using moving frames
[Une construction nouvelle des schémas invariants utilisant les repères mobiles]
Marx Chhay1  ; Aziz Hamdouni1
1 LEPTIAB, université de La Rochelle, avenue Michel-Crépeau, 17042 La Rochelle cedex 01, France
Comptes Rendus. Mécanique, Volume 338 (2010) no. 2, pp. 97-101.

On propose une procédure nouvelle de construction des repères mobiles permettant de rendre invariant les schémas de discrétisation en différences finies. Elle prend en compte l'ordre de consistance et garantit aux schémas invariants de meilleures performances que celles des schémas classiques. On illustre les performances de cette approche sur l'exemple de l'équation de Burgers.

We propose a new approach for moving frame construction that allows to make finite difference scheme invariant. This approach takes into account the order of accuracy and guarantees numerical properties of invariant schemes that overcome those of classical schemes. Benefits obtained with this process are illustrated with the Burgers equation.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2010.01.001
Keywords: Computational fluid mechanics, Invariant scheme, Lie symmetry, Moving frame, Finite difference scheme, Geometric integrator
Mots-clés : Mécanique des fluides numérique, Schéma invariant, Symétrie de Lie, Repères mobiles, Schéma aux différences finies, Intégrateur géométrique

Marx Chhay 1 ; Aziz Hamdouni 1

1 LEPTIAB, université de La Rochelle, avenue Michel-Crépeau, 17042 La Rochelle cedex 01, France 
@article{CRMECA_2010__338_2_97_0,
     author = {Marx Chhay and Aziz Hamdouni},
     title = {A new construction for invariant numerical schemes using moving frames},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {97--101},
     publisher = {Elsevier},
     volume = {338},
     number = {2},
     year = {2010},
     doi = {10.1016/j.crme.2010.01.001},
     language = {en},
}
TY  - JOUR
AU  - Marx Chhay
AU  - Aziz Hamdouni
TI  - A new construction for invariant numerical schemes using moving frames
JO  - Comptes Rendus. Mécanique
PY  - 2010
SP  - 97
EP  - 101
VL  - 338
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crme.2010.01.001
LA  - en
ID  - CRMECA_2010__338_2_97_0
ER  - 
%0 Journal Article
%A Marx Chhay
%A Aziz Hamdouni
%T A new construction for invariant numerical schemes using moving frames
%J Comptes Rendus. Mécanique
%D 2010
%P 97-101
%V 338
%N 2
%I Elsevier
%R 10.1016/j.crme.2010.01.001
%G en
%F CRMECA_2010__338_2_97_0
Marx Chhay; Aziz Hamdouni. A new construction for invariant numerical schemes using moving frames. Comptes Rendus. Mécanique, Volume 338 (2010) no. 2, pp. 97-101. doi : 10.1016/j.crme.2010.01.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.01.001/

[1] N.H. Ibragimov CRC Handbook of Lie Group Analysis of Differential Equations, vols. 1–3, CRC Press, Boca Raton, 1994

[2] P.J. Olver An introduction to moving frames (I.M. Mladenov; A.C. Hirschfeld, eds.), Geometry, Integrability and Quantization, vol. 5, Softex, Sofia, Bulgaria, 2004, pp. 67-80

[3] P. Kim Invariantization of the Crank–Nicolson method for burgers equation, Physica D: Nonlinear Phenomena, Volume 237 (2008) no. 2, pp. 243-254

[4] P.J. Olver Moving frames, J. Symb. Comp., Volume 3 (2003), pp. 501-512

[5] P.J. Olver Applications of Lie Groups to Differential Equations, Springer-Verlag, 1993

  • Géry de Saxcé; Dina Razafindralandy Lie groups and continuum mechanics: where do we stand today?, Comptes Rendus. Mécanique, Volume 351 (2024) no. S3, p. 135 | DOI:10.5802/crmeca.242
  • C. H. C. C. Basquerotto; A. Ruiz On the reduction of nonlinear mechanical systems via moving frames: a bead on a rotating wire hoop and a spinning top, Acta Mechanica, Volume 231 (2020) no. 12, p. 4867 | DOI:10.1007/s00707-020-02798-1
  • Cláudio H. C. Costa Basquerotto; Adrián Ruiz; Edison Righetto; Samuel da Silva Moving frames for Lie symmetries reduction of nonholonomic systems, Acta Mechanica, Volume 230 (2019) no. 8, p. 2963 | DOI:10.1007/s00707-019-02445-4
  • Dina Razafindralandy; Aziz Hamdouni; Marx Chhay A review of some geometric integrators, Advanced Modeling and Simulation in Engineering Sciences, Volume 5 (2018) no. 1 | DOI:10.1186/s40323-018-0110-y
  • Irvy M. A. Gledhill; Hamed Roohani; Karl Forsberg; Peter Eliasson; Beric W. Skews; Jan Nordström Theoretical treatment of fluid flow for accelerating bodies, Theoretical and Computational Fluid Dynamics, Volume 30 (2016) no. 5, p. 449 | DOI:10.1007/s00162-016-0382-0
  • Alexander Bihlo; Jean-Christophe Nave Convecting reference frames and invariant numerical models, Journal of Computational Physics, Volume 272 (2014), p. 656 | DOI:10.1016/j.jcp.2014.04.042
  • Elizabeth Mansfield; Gloria Marí Beffa; Jing Ping Wang Discrete Moving Frames and Discrete Integrable Systems, Foundations of Computational Mathematics, Volume 13 (2013) no. 4, p. 545 | DOI:10.1007/s10208-013-9153-0
  • Ying Li; Li Zhang; Jiaqiang Zhao; Yinghui Han, 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery (2012), p. 1716 | DOI:10.1109/fskd.2012.6234330
  • Alexander Bihlo; Roman O. Popovych Invariant Discretization Schemes for the Shallow-Water Equations, SIAM Journal on Scientific Computing, Volume 34 (2012) no. 6, p. B810 | DOI:10.1137/120861187
  • Elizabeth Mansfield; Jun Zhao On the Modern Notion of a Moving Frame, Guide to Geometric Algebra in Practice (2011), p. 411 | DOI:10.1007/978-0-85729-811-9_20
  • M. Chhay; E. Hoarau; A. Hamdouni; P. Sagaut Comparison of some Lie-symmetry-based integrators, Journal of Computational Physics, Volume 230 (2011) no. 5, p. 2174 | DOI:10.1016/j.jcp.2010.12.015
  • Marx Chhay; Aziz Hamdouni On the accuracy of invariant numerical schemes, Communications on Pure and Applied Analysis, Volume 10 (2010) no. 2, p. 761 | DOI:10.3934/cpaa.2011.10.761
  • Marx Chhay; Aziz Hamdouni Lie Symmetry Preservation by Finite Difference Schemes for the Burgers Equation, Symmetry, Volume 2 (2010) no. 2, p. 868 | DOI:10.3390/sym2020868

Cité par 13 documents. Sources : Crossref

Commentaires - Politique

Il n'y a aucun commentaire pour cet article. Soyez le premier à écrire un commentaire !

Publier un nouveau commentaire:

Publier une nouvelle réponse:

Ces articles pourraient vous intéresser

Lie groups and continuum mechanics: where do we stand today?

Géry de Saxcé; Dina Razafindralandy

C. R. Méca (2023)