An induced limitation in the reconstruction step for Euler equations of compressible gas dynamics in arbitrary dimension

Philippe Hoch

doi:10.5802/crmath.619

Article de recherche - Analyse numérique

An induced limitation in the reconstruction step for Euler equations of compressible gas dynamics in arbitrary dimension
[Une limitation induite pour l’étape de reconstruction dans les équations d’Euler en dimension quelconque]

Philippe Hoch ¹

¹ CEA-DAM, DIF, 91297, Arpajon Cedex, France

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 911-935.

Résumé
Abstract

On s’intéresse au processus de limitation pour la reconstruction des quantitées liées aux équations d’Euler de la dynamique des gaz compressibles pour une loi de pression générale de type $P (ρ, ϵ)$ (densité, énergie interne massique). Par exemple, pour la loi des gaz parfait, les contraintes sont $ρ > 0$ et $ϵ > 0$ , et la vitesse $U$ n’est a priori pas bornée dans le problème continu. Elle est néanmoins dans $L^{2} (Ω, ρ)$ comme conséquence de la relation sur les énergies $E = ϵ + \frac{1}{2} {| U |}^{2}$ dans $L^{1} (Ω, ρ)$ (par conservation globale de l’énergie totale $ρ E$ ). On montre un principe similaire dans le cadre d’une reconstruction conservative en dimension d’espace quelconque et pour un ordre de reconstruction lui aussi arbitraire. L’utilisation de la formule de Leibniz sur les variables massiques $ϵ$ , $U$ et $\frac{1}{2} {| U |}^{2}$ permet en effet d’obtenir en discret un contrôle induit de la vitesse reconstruite grâce au contrôle des reconstructions de la densité et des énergies. Nous construisons une limitation directe sur la variable poids $ρ$ et aussi surtout sur la variable massique $ϵ$ . En particulier, cette dernière permet de limiter, de manière induite, la vitesse $U$ . La reconstruction limitée des variables conservatives se déduit de l’assemblage de ces différents processus de limitation. Nous illustrons sur des cas tests, en dimension $d = 1$ et $d = 2$ , les reconstructions d’ordres 2 et 3 ainsi obtenues.

We are interested in the limitation process for the reconstruction of quantities related to Euler’s equations of compressible gas dynamics for a general pressure law of type $P (ρ, ϵ)$ (density, specific internal energy). For example, for perfect gas laws, we recall the constraints $ρ > 0$ and $ϵ > 0$ , and that the velocity $U$ is a priori not bounded in the continuous problem. Nevertheless it is in $L^{2} (Ω, ρ)$ as a consequence of the relation on the energies $E = ϵ + \frac{1}{2} {| U |}^{2}$ in $L^{1} (Ω, ρ)$ (due to global conservation of total energy $ρ E$ ). We show a similar result for conservative reconstruction in any space dimension and for an arbitrary reconstruction order. The use of the Leibniz formula on the specific variables $ϵ$ , $U$ and $\frac{1}{2} {| U |}^{2}$ allows to obtain also such a discrete induced control of reconstructed velocity thanks to control of reconstructed density and energies. We build a direct limitation on the weight variable $ρ$ and also especially on the specific variable $ϵ$ . In particular, the latter makes it possible to limit, in an induced way, the velocity $U$ . The limited reconstruction of the conservative variables is deduced from the assembly of these different limitation processes. We illustrate in dimension $d = 1$ and $d = 2$ on some test cases, our reconstructions of orders 2 and 3.

Reçu le : 2023-07-28
Révisé le : 2023-12-03
Accepté le : 2024-01-30
Publié le : 2024-10-07

DOI : 10.5802/crmath.619

Classification : 00X99
Keywords: Compressible Euler system, arbitrary order reconstruction, induced admissible limitation
Mot clés : Équations d’Euler compressibles, reconstruction d’ordre arbitraire, limitation induite admissible

Affiliations des auteurs :

Philippe Hoch ¹

¹ CEA-DAM, DIF, 91297, Arpajon Cedex, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Philippe Hoch},
     title = {An induced limitation in the reconstruction step for {Euler} equations of compressible gas dynamics in arbitrary dimension},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {911--935},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.619},
     language = {en},
}

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Philippe Hoch. An induced limitation in the reconstruction step for Euler equations of compressible gas dynamics in arbitrary dimension. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 911-935. doi : 10.5802/crmath.619. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.619/

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