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 (density, specific internal energy). For example, for perfect gas laws, we recall the constraints and , and that the velocity is a priori not bounded in the continuous problem. Nevertheless it is in as a consequence of the relation on the energies in (due to global conservation of total energy ). 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 , and 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 and also especially on the specific variable . In particular, the latter makes it possible to limit, in an induced way, the velocity . The limited reconstruction of the conservative variables is deduced from the assembly of these different limitation processes. We illustrate in dimension and on some test cases, our reconstructions of orders 2 and 3.
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 (densité, énergie interne massique). Par exemple, pour la loi des gaz parfait, les contraintes sont et , et la vitesse n’est a priori pas bornée dans le problème continu. Elle est néanmoins dans comme conséquence de la relation sur les énergies dans (par conservation globale de l’énergie totale ). 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 , et 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 et aussi surtout sur la variable massique . En particulier, cette dernière permet de limiter, de manière induite, la vitesse . 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 et , les reconstructions d’ordres 2 et 3 ainsi obtenues.
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Keywords: Compressible Euler system, arbitrary order reconstruction, induced admissible limitation
Mot clés : Équations d’Euler compressibles, reconstruction d’ordre arbitraire, limitation induite admissible
Philippe Hoch 1
@article{CRMATH_2024__362_G8_911_0, 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}, }
TY - JOUR AU - Philippe Hoch TI - An induced limitation in the reconstruction step for Euler equations of compressible gas dynamics in arbitrary dimension JO - Comptes Rendus. Mathématique PY - 2024 SP - 911 EP - 935 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.619 LA - en ID - CRMATH_2024__362_G8_911_0 ER -
%0 Journal Article %A Philippe Hoch %T An induced limitation in the reconstruction step for Euler equations of compressible gas dynamics in arbitrary dimension %J Comptes Rendus. Mathématique %D 2024 %P 911-935 %V 362 %I Académie des sciences, Paris %R 10.5802/crmath.619 %G en %F CRMATH_2024__362_G8_911_0
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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