The Leray-Gårding method for finite difference schemes

Coulombel, Jean-François

Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 297-331.

Dans les années 1950, Leray et Gårding ont développé une technique de multiplicateur pour obtenir des estimations a priori de solutions d’équations hyperboliques scalaires. L’existence d’un multiplicateur est le point de départ du travail de Rauch [23] pour montrer des estimations de semi-groupe pour les problèmes aux limites hyperboliques. Dans cet article, nous expliquons comment cette technique de multiplicateur peut être adaptée au cadre des schémas aux différences finies pour les équations de transport. Ce travail s’applique à des schémas numériques multi-pas en temps. L’existence et les propriétés du multiplicateur nous permettent d’obtenir des estimations de semi-groupe optimales pour des versions totalement discrètes des problèmes aux limites hyperboliques.

In the fifties, Leray and Gårding have developed a multiplier technique for deriving a priori estimates for solutions to scalar hyperbolic equations. The existence of such a multiplier is the starting point of the argument by Rauch [23] for the derivation of semigroup estimates for hyperbolic initial boundary value problems. In this article, we explain how this multiplier technique can be adapted to the framework of finite difference approximations of transport equations. The technique applies to numerical schemes with arbitrarily many time levels. The existence and properties of the multiplier enable us to derive optimal semigroup estimates for fully discrete hyperbolic initial boundary value problems.

Classification : 65M06, 65M12, 35L03, 35L04
Keywords: Hyperbolic equations, difference approximations, stability, boundary conditions, semigroup
Mot clés : Équations hyperboliques, différences finies, stabilité, conditions aux limites, semi-groupe

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