  We present a numerical method to solve the equations of general relativistic
hydrodynamics in a given external gravitational field. The method is based on a
generalization of Roe's approximate Riemann solver for the non relativistic
Euler equations in Cartesian coordinates. The new method is applied to a set of
standard test problems for general relativistic hydrodynamics, and is shown to
perform well in comparison to existing numerical schemes. In contrast to
existing explicit methods the present method can cope with strong relativistic
shocks. By-products are: the characteristic form of the general relativistic
Euler equations, a numerical method for special relativity that can deal with
strong discontinuities, a numerical scheme for the integration of the Euler
equations in an arbitrary coordinate system, possibly under the influence of
(external) gravity, and a novel method to incorporate source terms in numerical
schemes.
