  We study a finite volume scheme for the approximation of the solution to
convection diffusion equations with nonlinear convection and Robin boundary
conditions. The scheme builds on the interpretation of such a continuous
equation as the hydrodynamic limit of some simple exclusion jump process. We
show that the scheme admits a unique discrete solution, that the natural bounds
on the solution are preserved, and that it encodes the second principle of
thermodynamics in the sense that some free energy is dissipated along time. The
convergence of the scheme is then rigorously established thanks to compactness
arguments. Numerical simulations are finally provided, highlighting the overall
good behavior of the scheme.

Numerical Analysis;Numerical Analysis;Analysis of PDEs