  In this paper, we study the numerical stabilization of a 1D system of two
wave equations coupled by velocities with an internal, local control acting on
only one equation. In the theoretical part of this study, we distinguished two
cases. In the first one, the two waves assumed propagate at the same speed.
Under appropriate geometric conditions, we had proved that the energy decays
exponentially. While in the second case, when the waves propagate at different
speeds, under appropriate geometric conditions, we had proved that the energy
decays only at a polynomial rate. In this paper, we confirmed these two results
in a 1D numerical approximation. However, when the coupling region does not
intersect the damping region, the stabilization of the system is still
theoretically an open problem. But, here in both cases, we observed an
unpredicted behavior : the energy decays at an exponential rate when the
propagation speeds are the same or at a polynomial rate when they are
different.
