  A numerical study is presented to analyze the thermal mechanisms of unsteady,
supersonic granular flow, by means of hydrodynamic simulations of the
Navier-Stokes granular equations. For this purpose a paradigmatic problem in
granular dynamics such as the Faraday instability is selected. Two different
approaches for the Navier-Stokes transport coefficients for granular materials
are considered, namely the traditional Jenkins-Richman theory for moderately
dense quasi-elastic grains, and the improved Garz\'o-Dufty-Lutsko theory for
arbitrary inelasticity, which we also present here. Both solutions are compared
with event-driven simulations of the same system under the same conditions, by
analyzing the density, the temperature and the velocity field. Important
differences are found between the two approaches leading to interesting
implications. In particular, the heat transfer mechanism coupled to the density
gradient which is a distinctive feature of inelastic granular gases, is
responsible for a major discrepancy in the temperature field and hence in the
diffusion mechanisms.

Fluid Dynamics;Soft Condensed Matter