  In this paper we explore numerically the evolution of a warped accretion
disc. Here, we focus here on the regime where the warp evolves diffusively. By
comparing the numerical results to a simple diffusion model, we are able to
determine the diffusion coefficient of the warp, $\alpha_2$, as a function of
the relevant disc parameters. We find that while in general the disc behaviour
is well reproduced by the diffusion model and for relatively large viscosities
the warp diffusion is well described by the linear theory (in particular
confirming that the warp diffusion coefficient is inversely proportional to
viscosity), significant non-linear effects are present as the viscosity becomes
smaller, but still dominates over wave-propagation effects. In particular, we
find that the inverse dependence of the diffusion coefficient on viscosity
breaks down at low viscosities, so that $\alpha_2$ never becomes larger than a
saturation value $\alpha_{\rm max}$ of order unity. This can have major
consequences in the evolution of systems where a warped disc is present. In
particular, it affects the location of the warp radius in the Bardeen-Petterson
effect and therefore the spin up (or spin down) of supermassive black holes in
the nuclei of galaxies. Additionally, we also find that while the rate of warp
diffusion does not depend significantly on the detailed viscosity formulation,
the rate of internal precession generated by the warp is strongly affected by
it. Such effects should be considered with care when modeling the evolution of
warped discs. This emphasises the need to test the above results using
different numerical schemes, and with higher resolution, in order to
investigate the degree to which numerical simulations are able to provide
accurate modeling of the complex fluid dynamics of warped discs. (Abridged)
