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Lie group study of finite difference schemes

Pages: 495 - 505, Issue Special, September 2007

 Abstract        Full Text (384.7K)              

Emma Hoarau - Onera, Computational Fluid Dynamics and Aeroacoustics Department (DSNA), BP 72, 20 avenue de la Division Leclerc, 92322 Châtillon Cedex, France (email)
Claire david@lmm.jussieu.fr David - Université Pierre et Marie Curie-Paris 6, Institut Jean Le Rond d'Alembert, UMR CNRS 71900, Boîte courrier $n^0$ 162, 4 place Jussieu, 75252 Paris, cedex 05. France, France (email)
Pierre Sagaut - Université Pierre et Marie Curie-Paris 6, Institut Jean Le Rond d'Alembert, UMR CNRS 71900, Boîte courrier $n^0$ 162, 4 place Jussieu, 75252 Paris, cedex 05. France, France (email)
Thiên-Hiêp Lê - ONERA, Computational Fluid Dynamics and Aeroacoustics Department (DSNA), BP 72, 20 avenue de la Division Leclerc, 92322 Châtillon Cedex, France (email)

Abstract: Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference approximation, and lead to inaccurate numerical results. This paper deals with the analysis of symmetry group of finite difference equations, which is based on the differential approximation. We develop a new scheme, the related differential approximation of which is invariant under the symmetries of the original differential equations. A comparison of numerical performance of this scheme, with standard ones and a higher order one has been realized for the Burgers equation.

Keywords:  Lie group; semi-invariant scheme; continuous symmetry; differential approximation; finite difference scheme; artificial viscosity term.
Mathematics Subject Classification:  22E70.

Received: September 2006;      Revised: August 2007;      Published: September 2007.