  The stability theory for hyperbolic initial boundary value problems relies
most of the time on the Laplace transform with respect to the time variable.
For technical reasons, this usually restricts the validity of stability
estimates to the case of zero initial data. In this article, we consider the
class of non-glancing finite difference approximations to the hyperbolic
operator. We show that the maximal stability estimates that are known for zero
initial data and nonzero boundary source term extend to the case of nonzero
initial data in \^a 2 . The main novelty of our approach is to cover finite
difference schemes with an arbitrary number of time levels. As an easy
corollary of our main trace estimate, we recover former stability results in
the semigroup sense by Kreiss [Kre68] and Osher [Osh69b].
