  In this article, we give a unified theory for constructing boundary layer
expansions for dis-cretized transport equations with homogeneous Dirichlet
boundary conditions. We exhibit a natural assumption on the discretization
under which the numerical solution can be written approximately as a two-scale
boundary layer expansion. In particular, this expansion yields discrete
semigroup estimates that are compatible with the continuous semigroup estimates
in the limit where the space and time steps tend to zero. The novelty of our
approach is to cover numerical schemes with arbitrarily many time levels, while
semigroup estimates were restricted, up to now, to numerical schemes with two
time levels only.
