LIE GROUP STUDY OF FINITE DIFFERENCE SCHEMES

Emma Hoarau (1) , Claire David (2) , Pierre Sagaut (2) , Thiên-Hiêp Lê (1)

1 ONERA - The French Aerospace Lab [Châtillon]
2 UPMC - Université Pierre et Marie Curie - Paris 6

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. 1. Introduction. Lie groups were introduced by Sophus Lie in 1870 in order to study the symmetries of differential equations, yielding thus analytical solutions. Literature provides substantial works and applications, [3], [4]. Symmetry groups can be determined by an automatic procedure, but it often turns out to be tedious and induces errors. A large amount of packages using symbolic manipulations of mathematical expressions have been written. We mention here some of those works: Schwartz [17], Vu and Carminati[14], Herod [15], Baumann [16], Cantwell [5]. In this paper we are interested in the application of the theory of Lie group to numerical analysis. Finite difference equations used to approximate the solutions of a differential equation generally do not respect the symmetries of the original equation, and can lead to inaccurate numerical results. Various techniques, that enable us to build a scheme preserving the symmetries of the original differential equation, have been studied. One of these techniques consists in constructing an invariant scheme from a given one by applying the method of the moving frame in [7], [8]. Another one consists in constructing an invariant scheme with the help of the discret invariants of its symmetry group [9], [10], [11], [12], [13] and provides the building of symmetry-adapted meshes, in preserving the differential equation symmetries. This technique is based on a direct study of the symmetries of difference equations and lattices. Yanenko [2] and Shokin [1], proposed to apply the Lie group theory to finite difference equations by means of the differential approximation. Thus, they have set 2000 Mathematics Subject Classification. 22E70.

Mots clés: and phrases Lie group; semi-invariant scheme; continuous symmetry; finite difference scheme; artificial viscosity term; differential approximation

Domaines: Analyse numérique [math.NA]; Sciences de l'ingénieur [physics]

Emma Hoarau, Claire David, Pierre Sagaut, Thiên-Hiêp Lê. LIE GROUP STUDY OF FINITE DIFFERENCE SCHEMES. Discrete and Continuous Dynamical Systems - Series B, 2007, 2007 (speciale), pp.495 - 505. ⟨10.3934/proc.2007.2007.495⟩. ⟨hal-00116811v2⟩