  The aim of this article is to propose a systematic study of transparent
boundary conditions for finite difference approximations of evolution
equations. We try to keep the discussion at the highest level of generality in
order to apply the theory to the broadest class of problems. We deal with two
main issues. We first derive transparent numerical boundary conditions, that
is, we exhibit the relations satisfied by the solution to the pure Cauchy
problem when the initial condition vanishes outside of some domain. Our
derivation encompasses discretized transport, diffusion and dispersive
equations with arbitrarily wide stencils. The second issue is to prove sharp
stability estimates for the initial boundary value problem obtained by
enforcing the boundary conditions derived in the first step. We focus here on
discretized transport equations. Under the assumption that the numerical
boundary is non-characteristic, our main result characterizes the class of
numerical schemes for which the corresponding transparent boundary conditions
satisfy the so-called Uniform Kreiss-Lopatinskii Condition introduced in
[GKS72]. Adapting some previous works to the non-local boundary conditions
considered here, our analysis culminates in the derivation of trace and
semigroup estimates for such transparent numerical boundary conditions. Several
examples and possible extensions are given.
