[Théorème $H$ discret pour une discrétisation volumes finis d’un système cinétique non-linéaire : application à l’hypocoercivité]
In this article, we study the long-time behavior of a finite-volume discretization for a nonlinear kinetic reaction model involving two interacting species. Building upon the seminal work of Favre, Pirner, Schmeiser [Arch. Ration. Mech. Anal. 247 (2023), no. 4], we extend the discrete exponential convergence to equilibrium result established in Bessemoulin-Chatard, Laidin, Rey [IMA J. Numer. Anal. 45 (2025), no. 4], which was obtained in a perturbative framework using weighted $L^2$ estimates. The analysis applies to a broader class of exponentially decaying initial data, without requiring proximity to equilibrium, by exploiting the properties of the Boltzmann entropy. The proof relies on the propagation of the initial $L^\infty $ bounds, derived from monotonicity properties of the scheme, allowing controlled linearizations within the nonlinear entropy estimates. Moreover, we show that the time-discrete dissipation inherent to the numerical scheme plays a crucial stabilizing role, providing control over the nonlinear terms.
Dans cet article, nous étudions le comportement en temps long d’une discrétisation volumes finis pour un modèle cinétique de réaction non linéaire faisant intervenir deux espèces en interaction. En nous appuyant sur le travail de Favre, Pirner, Schmeiser [Arch. Ration. Mech. Anal. 247 (2023), no. 4], nous étendons le résultat de convergence exponentielle discret vers l’équilibre, établi dans Bessemoulin-Chatard, Laidin, Rey [IMA J. Numer. Anal. 45 (2025), no. 4], obtenu dans un cadre perturbatif à l’aide d’estimations en norme $L^2$ pondérée. L’analyse s’applique à une classe plus large de données initiales décroissant exponentiellement, sans imposer de proximité à l’équilibre, en exploitant les propriétés de l’entropie de Boltzmann. La preuve repose sur la propagation des bornes $L^\infty $, obtenues grâce aux propriétés de monotonie du schéma, permettant des linéarisations contrôlées au sein des estimations d’entropie non linéaires. De plus, nous montrons que la dissipation discrète en temps inhérente au schéma numérique joue un rôle stabilisant crucial, fournissant un contrôle des termes non linéaires.
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Keywords: Kinetic equations, hypocoercivity, maximum principle, finite volume methods, large time behavior
Mots-clés : Équations cinétiques, hypocoercivité, principe du maximum, méthodes volumes finis, comportement en temps long
Marianne Bessemoulin-Chatard 1 ; Tino Laidin 2 ; Thomas Rey 3
CC-BY 4.0
Marianne Bessemoulin-Chatard; Tino Laidin; Thomas Rey. Discrete $H$-theorem for finite volume discretization of a nonlinear kinetic system: application to hypocoercivity. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 635-664. doi: 10.5802/crmath.855
@article{CRMATH_2026__364_G3_635_0,
author = {Marianne Bessemoulin-Chatard and Tino Laidin and Thomas Rey},
title = {Discrete $H$-theorem for finite volume discretization of a nonlinear kinetic system: application to hypocoercivity},
journal = {Comptes Rendus. Math\'ematique},
pages = {635--664},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {364},
doi = {10.5802/crmath.855},
language = {en},
}
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