A semi-implicit method for hyperbolic problems with different time-scales

Hyperbolic systems with two different time-scales are considered, where the solutions vary on the slow scale only. For this type of problem semi-implicit difference methods are very natural, and in this paper we analyze the leap-frog backwards Euler scheme. In particular it is shown, that when the ratio $\varepsilon $ between the slow and the fast scale tends to zero, the solutions of the approximation converge to solutions of the reduced differential equation. Numerical experiments are included for illustration of the theoretical results.