We study numerical approximations of systems of partial differential
equations modeling the interaction of short and long waves. The short waves are
modeled by a nonlinear Schr\"odinger equation which is coupled to another
equation modeling the long waves. Here, we consider the case where the long
wave equation is either a hyperbolic conservation law or a Korteweg--de Vries
equation. In the former case, we prove the strong convergence of a
Lax--Friedrichs type scheme towards the unique entropy solution of the problem,
while in the latter case we prove convergence of a finite difference scheme
towards the global solution of the problem.