Abstract
Since the celebrated theorem of Lax and Wendroff, we know a necessary condition that any numerical scheme for hyperbolic problem should satisfy: it should be written in flux form. A variant can also be formulated for the entropy. Even though some schemes, as for example those using continuous finite element, do not formally cast into this framework, it is a very convenient one. In this paper, we revisit this, introduce a different notion of local conservation which contains the previous one in one space dimension, and explore its consequences. This gives a more flexible framework that allows to get, systematically, entropy stable schemes, entropy dissipative ones, or accommodate more constraints. In particular, we can show that continuous finite element method can be rewritten in the finite volume framework, and all the quantities involved are explicitly computable. We end by presenting the only counter example we are aware of, i.e a scheme that seems not to be rewritten as a finite volume scheme.
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Notes
- 1.
If the set S is discrete, \(\vert S\vert \) is its cardinal. If S is part of a domain, it is its measure with respect to the lebesgue measure, i.e. its length/area/volume.
- 2.
This is the essence of Roe’s 1981 paper: setting \(\varPhi _i^{[x_i, x_{i+1}]}=a^-_{i+1/2}(\textbf{u}_{i+1}-\textbf{u}_i)\) and \(\varPhi _{i+1}^{[x_i, x_{i+1}]}=a_{j+1/2}^+(\textbf{u}_{i+1}-\textbf{u}_i)\), we see that the method of characteristics (8) is conservative if and only if \({\textbf{f}}(\textbf{u}_{i+1})-{\textbf{f}}(\textbf{u}_i)=a_{i+1/2}(\textbf{u}_{i+1}-\textbf{u}_i)\).
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Acknowledgements
I take this opportunity to thanks my many collaborators and students: M. Ricchiuto (Inria), P.H. Maire (Cea), F. Vilar (Montpellier), R. Loubère (Bordeaux), Ph. Öffner (Mainz), W. Barsukow (Bordeaux), M. Dumbser (Trento), S. Busto (Vigò), S. Tokareva (Los Alamos), P. Baccigaluppi (Milano), L. Micalizzi (Zürich), A. Larat (Grenoble), M. Mezine (Numeca), M. Lukakova (Mainz). Without them nothing would have been possible.
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Abgrall, R. (2023). A Personal Discussion on Conservation, and How to Formulate It. In: Franck, E., Fuhrmann, J., Michel-Dansac, V., Navoret, L. (eds) Finite Volumes for Complex Applications X—Volume 1, Elliptic and Parabolic Problems. FVCA 2023. Springer Proceedings in Mathematics & Statistics, vol 432. Springer, Cham. https://doi.org/10.1007/978-3-031-40864-9_1
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