Discrete $H$-theorem for a finite volume discretization of a nonlinear kinetic system: application to hypocoercivity
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| Format: | Preprint |
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2025
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| _version_ | 1866909908477149184 |
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| author | Bessemoulin-Chatard, Marianne Laidin, Tino Rey, Thomas |
| author_facet | Bessemoulin-Chatard, Marianne Laidin, Tino Rey, Thomas |
| contents | 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, ARMA, 2023], we extend the discrete exponential convergence to equilibrium result established in [Bessemoulin-Chatard, Laidin, Rey, IMAJNA, 2025], 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_13323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete $H$-theorem for a finite volume discretization of a nonlinear kinetic system: application to hypocoercivity Bessemoulin-Chatard, Marianne Laidin, Tino Rey, Thomas Numerical Analysis 82B40, 65M08, 65M12 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, ARMA, 2023], we extend the discrete exponential convergence to equilibrium result established in [Bessemoulin-Chatard, Laidin, Rey, IMAJNA, 2025], 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. |
| title | Discrete $H$-theorem for a finite volume discretization of a nonlinear kinetic system: application to hypocoercivity |
| topic | Numerical Analysis 82B40, 65M08, 65M12 |
| url | https://arxiv.org/abs/2511.13323 |