  We study the onset of patterns in vertically oscillated layers of
frictionless dissipative particles. Using both numerical solutions of continuum
equations to Navier-Stokes order and molecular dynamics (MD) simulations, we
find that standing waves form stripe patterns above a critical acceleration of
the cell. Changing the frequency of oscillation of the cell changes the
wavelength of the resulting pattern; MD and continuum simulations both yield
wavelengths in accord with previous experimental results. The value of the
critical acceleration for ordered standing waves is approximately 10% higher in
molecular dynamics simulations than in the continuum simulations, and the
amplitude of the waves differs significantly between the models. The delay in
the onset of order in molecular dynamics simulations and the amplitude of noise
below this onset are consistent with the presence of fluctuations which are
absent in the continuum theory. The strength of the noise obtained by fit to
Swift-Hohenberg theory is orders of magnitude larger than the thermal noise in
fluid convection experiments, and is comparable to the noise found in
experiments with oscillated granular layers and in recent fluid experiments on
fluids near the critical point. Good agreement is found between the mean field
value of onset from the Swift-Hohenberg fit and the onset in continuum
simulations. Patterns are compared in cells oscillated at two different
frequencies in MD; the layer with larger wavelength patterns has less noise
than the layer with smaller wavelength patterns.
