Error Bounded Schemes for Time-dependent Hyperbolic Problems

In this paper we address the error growth in time for hyperbolic problems on first order form. The energy method is used to study when an error growth or a fixed error bound is obtained. It is shown that the choice of boundary procedure is a crucial point. Numerical experiments corroborate the theoretical findings. 1. Introduction. Stable approximations of hyperbolic problems on first order form often exhibit a linear (or near linear) error growth in time, see for example [15],[13],[6]. In many of those cases, the wave is restricted or trapped in the domain for long times. Typical examples include periodic problems or problems where the wave is trapped in cavities. However, also cases with a definite bound on the error as time passes can be observed. Typically, in such cases, the wave propagates through the domain for a limited time as in an inflow-outflow problem. To initiate our investigation we consider the problem