  We introduce a formulation of Eulerian general relativistic hydrodynamics
which is applicable for (perfect) fluid data prescribed on either spacelike or
null hypersurfaces. Simple explicit expressions for the characteristic speeds
and fields are derived in the general case. A complete implementation of the
formalism is developed in the case of spherical symmetry. The algorithm is
tested in a number of different situations, predisposing for a range of
possible applications. We consider the Riemann problem for a polytropic gas,
with initial data given on a retarded/advanced time slice of Minkowski
spacetime. We compute perfect fluid accretion onto a Schwarzschild black hole
spacetime using ingoing null Eddington-Finkelstein coordinates. Tests of fluid
evolution on dynamic background include constant density and TOV stars sliced
along the radial null cones. Finally, we consider the accretion of
self-gravitating matter onto a central black hole and the ensuing increase in
the mass of the black hole horizon.
