  In this article we model a Global Navigation Satellite System (GNSS) in a
Schwarzschild space-time, as a first approximation of the relativistic geometry
around the Earth. The closed time-like and scattering light-like geodesics are
obtained analytically, describing respectively trajectories of satellites and
electromagnetic signals. We implement an algorithm to calculate Schwarzschild
coordinates of a GNSS user who receives proper times sent by four satellites,
knowing their orbital parameters; the inverse procedure is implemented to check
for consistency. The constellation of satellites therefore realizes a
geocentric inertial reference system with no \emph{a priori} realization of a
terrestrial reference frame. We show that the calculation is very fast and
could be implemented in a real GNSS, as an alternative to usual post-Newtonian
corrections. Effects of non-gravitational perturbations on positioning errors
are assessed, and methods to reduce them are sketched. In particular,
inter-links between satellites could greatly enhance stability and accuracy of
the positioning system.
