  We present a numerical implementation of radiative transfer based on an
explicitly photon-conserving advection scheme, where radiative fluxes over the
cell interfaces of a structured or unstructured mesh are calculated with a
second-order reconstruction of the intensity field. The approach employs a
direct discretisation of the radiative transfer equation in Boltzmann form with
adjustable angular resolution that in principle works equally well in the
optically thin and optically thick regimes. In our most general formulation of
the scheme, the local radiation field is decomposed into a linear sum of
directional bins of equal solid-angle, tessellating the unit sphere. Each of
these "cone-fields" is transported independently, with constant intensity as a
function of direction within the cone. Photons propagate at the speed of light
(or optionally using a reduced speed of light approximation to allow larger
timesteps), yielding a fully time-dependent solution of the radiative transfer
equation that can naturally cope with an arbitrary number of sources, as well
as with scattering. The method casts sharp shadows, subject to the limitations
induced by the adopted angular resolution. If the number of point sources is
small and scattering is unimportant, our implementation can alternatively treat
each source exactly in angular space, producing shadows whose sharpness is only
limited by the grid resolution. A third hybrid alternative is to treat only a
small number of the locally most luminous point sources explicitly, with the
rest of the radiation intensity followed in a radiative diffusion
approximation. We have implemented the method in the moving-mesh code {\small
AREPO}, where it is coupled to the hydrodynamics in an operator splitting
approach that subcycles the radiative transfer alternatingly with the
hydrodynamical evolution steps.
