  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.
