  Understanding the velocity field is very important for modern cosmology: it
gives insights to structure formation in general, and also its properties are
crucial ingredients in modelling redshift-space distortions and in interpreting
measurements of the kinetic Sunyaev-Zeldovich effect. Unfortunately,
characterising the velocity field in cosmological N-body simulations is
inherently complicated by two facts: i) The velocity field becomes manifestly
multi-valued after shell-crossing and has discontinuities at caustics. This is
due to the collisionless nature of dark matter. ii) N-body simulations sample
the velocity field only at a set of discrete locations, with poor resolution in
low-density regions. In this paper, we discuss how the associated problems can
be circumvented by using a phase-space interpolation technique. This method
provides extremely accurate estimates of the cosmic velocity fields and its
derivatives, which can be properly defined without the need of the arbitrary
"coarse-graining" procedure commonly used. We explore in detail the
configuration-space properties of the cosmic velocity field on very large
scales and in the highly nonlinear regime. In particular, we characterise the
divergence and curl of the velocity field, present their one-point statistics,
analyse the Fourier-space properties and provide fitting formulae for the
velocity divergence bias relative to the non-linear matter power spectrum. We
furthermore contrast some of the interesting differences in the velocity fields
of warm and cold dark matter models. We anticipate that the high-precision
measurements carried out here will help to understand in detail the dynamics of
dark matter and the structures it forms.
