  We present two continuum models A and B to study the convective instability
of granular materials subjected to vibrations. We carry out the linear
stability analysis for model A and uncover the instability mechanism as a
supercritical bifurcation of a bouncing solution. We also explicitly determine
the onset of convection as a function of control parameters. The results of
simulations are in excellent agreement with the stability analysis. Additional
feature of the model B is the inclusion of the relaxation term in the momentum
equation, which appears to be crucial in capturing what is missing in model A,
in particular, in reproducing experimental convection patterns for large aspect
ratio, both vertically, in which case convective rolls move toward the surface,
and horizontally, in which case convective rolls survive near the wall but are
suppressed in the bulk region.
