  We perform axisymmetric hydrodynamical simulations that describe the
nonlinear outcome of the viscous overstability in dense planetary rings. These
simulations are particularly relevant for Cassini observations of fine-scale
structure in Saturn's A and B-ring, which take the form of periodic
microstructure on the 0.1 km scale, and irregular larger-scale structure on
1-10 km. Nonlinear wavetrains dominate all the simulations, and we associate
them with the observed periodic microstructure. The waves can undergo small
chaotic fluctuations in their phase and amplitude, and may be punctuated by
more formidable `wave defects' distributed on longer scales. It is unclear,
however, whether the defects are connected to the irregular larger-scale
variations observed by Cassini. The long-term behaviour of the simulations is
dominated by the imposed boundary conditions, and more generally by the
limitations of the local model we use: the shearing box. When periodic boundary
conditions are imposed, the system eventually settles on a uniform travelling
wave of a predictable wavelength, while reflecting boundaries, and boundaries
with buffer zones, maintain a disordered state. The simulations omit
self-gravity, though we examine its influence in future work.
