In this article, we propose a finite volume discretization of a one dimensional nonlinear reaction kinetic model proposed in [Neumann, Schmeiser, Kint. Rel. Mod. 2016], which describes a 2-species recombination-generation process. Specifically, we establish the long-time convergence of approximate solutions towards equilibrium, at exponential rate. The study is based on an adaptation for a discretization of the linearized problem of the $L^2$ hypocoercivity method introduced in [Dolbeault, Mouhot, Schmeiser, 2015]. From this, we can deduce a local result for the discrete nonlinear problem. As in the continuous framework, this result requires the establishment of a maximum principle, which necessitates the use of monotone numerical fluxes.

翻译：本文针对[Neumann, Schmeiser, Kint. Rel. Mod. 2016]提出的一维非线性反应动理学模型（描述双物种重组-生成过程），提出其有限体积离散格式。具体地，我们建立了近似解以指数速率向平衡态长期收敛的特性。该研究基于对[Dolbeault, Mouhot, Schmeiser, 2015]提出的$L^2$次压缩性方法在离散线性化问题中的适应性修改。由此，我们可推导出离散非线性问题的局部结果。与连续框架类似，该结果需建立最大值原理，这要求使用单调数值通量。