  Thinking of flows crossing through regular porous media, we numerically
explore the behavior of weak solutions to a two-scale elliptic-parabolic system
that is strongly coupled by means of a suitable nonlinear dispersion term. The
two-scale system of interest originates from the fast-drift periodic
homogenization of a nonlinear convective-diffusion-reaction problem, where the
structure of the non-linearity in the drift fits to the hydrodynamic limit of a
totally asymmetric simple exclusion process for a population of particles. In
this article, we focus exclusively on numerical simulations that employ two
decoupled approximation schemes, viz. 'scheme 1' - a Picard-type iteration -
and 'scheme 2' - a time discretization decoupling. Additionally, we describe a
computational strategy which helps to drastically improve computation times.
Finally, we provide several numerical experiments to illustrate what dispersion
effects are introduced by a specific choice of microstructure and model
ingredients.
