Shocks in Vertically Oscillated Granular Layers
  We study shock formation in vertically oscillated granular layers, using both molecular dynamics simulations and numerical solutions of continuum equations to Navier-Stokes order. A flat layer of grains is thrown up from an oscillating plate during each oscillation cycle and collides with the plate later in the cycle. The collisions produce layer compaction near the plate and a high temperature shock front that rapidly propagates upward through the layer. The shock is highly time-dependent, propagating through the layer in only a quarter of the cycle. We compare numerical solutions of the continuum equations to molecular dynamics simulations that assume binary, instantaneous collisions between frictionless hard spheres. The two simulations yield results for the shock position, shape, and speed that agree well. An investigation of the effect of inelasticity shows that the shock velocity increases continuously with decreasing inelasticity; the elastic limit is not singular. 
J. Bougie, Sung Joon Moon, J. B. Swift, and Harry L. Swinney