The Excitation, Propagation and Dissipation of Waves in Accretion Discs:   The Non-linear Axisymmetric Case
  We analyse the non-linear propagation and dissipation of axisymmetric waves in accretion discs using the ZEUS-2D hydrodynamics code. The waves are numerically resolved in the vertical and radial directions. Both vertically isothermal and thermally stratified accretion discs are considered. The waves are generated by means of resonant forcing and several forms of forcing are considered. Compressional motions are taken to be locally adiabatic ($\gamma = 5/3$). Prior to non-linear dissipation, the numerical results are in excellent agreement with the linear theory of wave channelling in predicting the types of modes that are excited, the energy flux by carried by each mode, and the vertical wave energy distribution as a function of radius. In all cases, waves are excited that propagate on both sides of the resonance (inwards and outwards). For vertically isothermal discs, non-linear dissipation occurs primarily through shocks that result from the classical steepening of acoustic waves. For discs that are substantially thermally stratified, wave channelling is the primary mechanism for shock generation. Wave channelling boosts the Mach number of the wave by vertically confining the wave to a small cool region at the base of the disc atmosphere. In general, outwardly propagating waves with Mach numbers near resonance ${\cal M}_{\rm r} \ga 0.01$ undergo shocks within a distance of order the resonance radius. 
M. R. Bate, G. I. Ogilvie, S. H. Lubow, and J. E. Pringle