An induced limitation in the reconstruction step for Euler equations of compressible gas dynamics in arbitrary dimension
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 $P(\rho ,\epsilon )$ (density, specific internal energy). For example, for perfect gas laws, we recall the constraints $\rho &...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-10-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.619/ |
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| Summary: | 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 $P(\rho ,\epsilon )$ (density, specific internal energy). For example, for perfect gas laws, we recall the constraints $\rho >0$ and $\epsilon >0$, and that the velocity $\mathbf{U}$ is a priori not bounded in the continuous problem. Nevertheless it is in $L^2(\Omega ,\rho )$ as a consequence of the relation on the energies $E=\epsilon + \frac{1}{2} |\mathbf{U}|^2$ in $L^1(\Omega ,\rho )$ (due to global conservation of total energy $\rho E$). 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 $\epsilon $, $\mathbf{U}$ and $\frac{1}{2}|\mathbf{U}|^2$ 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 $\rho $ and also especially on the specific variable $\epsilon $. In particular, the latter makes it possible to limit, in an induced way, the velocity $\mathbf{U}$. 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. |
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| ISSN: | 1778-3569 |