  We take further steps in the development of the characteristic approach to
enable handling the physical problem of a compact self-gravitating object, such
as a neutron star, in close orbit around a black hole. We examine different
options for setting the initial data for this problem and, in order to shed
light on their physical relevance, we carry out short time evolution of this
data. To this end we express the matter part of the characteristic gravity code
so that the hydrodynamics are in conservation form. The resulting gravity plus
matter relativity code provides a starting point for more refined future
efforts at longer term evolution. In the present work we find that,
independently of the details of the initial gravitational data, the system
quickly flushes out spurious gravitational radiation and relaxes to a
quasi-equilibrium state with an approximate helical symmetry corresponding to
the circular orbit of the star.
