An induced limitation in the reconstruction step for Euler equations of compressible gas dynamics in arbitrary dimension
Year: 2024
Type: article
Abstract: 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 + 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 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.
Source: Comptes Rendus Mathématique
DOI: 10.5802/crmath.619
Author: Philippe Hoch
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4353028693