  We present the results of a study of propagating warp or bending waves in
accretion discs. Three dimensional hydrodynamic simulations were performed
using SPH, and the results of these are compared with calculations based on the
linear theory of warped discs. We consider primarily the physical regime in
which the dimensionless viscosity parameter `alpha' < H/r, the disc aspect
ratio, so that bending waves are expected to propagate. We also present
calculations in which `alpha' > H/r, where the warps are expected to behave
diffusively. Small amplitude perturbations are studied in both Keplerian and
slightly non Keplerian discs, and we find that the SPH results can be
reasonably well fitted by those of the linear theory. The main results of these
calculations are: (1) the warp in Keplerian discs when `alpha' < H/r propagates
with little dispersion and damps at a rate expected from estimates of the code
viscosity, (2) warps evolve diffusively when `alpha' > H/r, (3) the non
Keplerian discs exhibit a substantially more dispersive behaviour of the warps.
Initially imposed higher amplitude nonlinear warping disturbances were studied
in Keplerian discs. The results indicate that nonlinear warps can lead to the
formation of shocks, and that the evolution of the warp becomes less wave-like
and more diffusive in character. This work is relevant to the study of the
warped accretion discs that may occur around Kerr black holes or in misaligned
binary systems. The results indicate that SPH can accurately model the
hydrodynamics of warped discs, even when using rather modest numbers of
particles.
