A new construction for invariant numerical schemes using moving frames

Marx Chhay; Aziz Hamdouni

doi:10.1016/j.crme.2010.01.001

A new construction for invariant numerical schemes using moving frames

Marx Chhay ¹ ; Aziz Hamdouni ¹

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.

Abstract
Résumé

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.

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.

Received: 2009-01-27
Accepted: 2010-01-21
Published online: 2010-02-01

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

Author's affiliations:

Marx Chhay ¹; Aziz Hamdouni ¹

¹ LEPTIAB, université de La Rochelle, avenue Michel-Crépeau, 17042 La Rochelle cedex 01, France

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     author = {Marx Chhay and Aziz Hamdouni},
     title = {A new construction for invariant numerical schemes using moving frames},
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     pages = {97--101},
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     volume = {338},
     number = {2},
     year = {2010},
     doi = {10.1016/j.crme.2010.01.001},
     language = {en},
}

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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/

References
Cited by

[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

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