  We introduce a generalization of Glimm's random choice method, which provides
us with an approximation of entropy solutions to quasilinear hyperbolic system
of balance laws. The flux-function and the source term of the equations may
depend on the unknown as well as on the time and space variables. The method is
based on local approximate solutions of the generalized Riemann problem, which
form building blocks in our scheme and allow us to take into account naturally
the effects of the flux and source terms. To establish the nonlinear stability
of these approximations, we investigate nonlinear interactions between
generalized wave patterns. This analysis leads us to a global existence result
for quasilinear hyperbolic systems with source-term, and applies, for instance,
to the compressible Euler equations in general geometries and to hyperbolic
systems posed on a Lorentzian manifold.
