The effective use of finite difference schemes for evolution problems requires a specific treatment at the boundaries so as to truncate artificially the computational domain and/or to incorporate properly the modeled boundary conditions. The stability and consistency properties that are specific to this boundary treatment play a crucial role for the applicability of the whole scheme. Typically, parasitic boundary modes are likely to develop and may reduce its accuracy or even damage its stability. The two-scale expansions of the approximation solution is the most expedient way to understand the influence of such difficulties. As an example, the talk will deal with the simplest transport equation with outflow boundary conditions solved using multistep schemes, first in the case of homogeneous Dirichlet boundary conditions, and then for higher order numerical Neumann boundary condition. This is a joint work with Jean-François Coulombel, extended during the last CEMRACS within a project supported by the ANR project NABUCO (ANR-17-CE40-0025).