This habilitation manuscript gathers the work I have done in recent years. They mainly focus on the study of the stability and the multiscale analysis of finite difference methods for the approximation of linear hyperbolic problems with boundaries. In such a context, various scales are likely to be present, related for example to the phenomena of viscosity, relaxation or discretization. Then, the interactions at these scales between the interior problem and the boundary of the computational domain are liable for unexpected parasitic effects, such as boundary layers. They often severely impair the stability properties in the asymptotic process and sometimes reduce the quality and the accuracy of the approximation. Therefore, it appears crucial to discriminate and rule out pathological situations. The first three chapters relate successively to 1) the general theory of stability for the discrete problem in a bounded domain and the numerical verification of the discrete uniform Kreiss-Lopatinskii condition, 2) the construction and the use of asymptotic multi-scale expansions for the consistency analysis at the boundary, and 3) the uniform character of boundary stability properties in the presence of a relaxation limit. The next two chapters deal with geometric aspects in the large-time asymptotic of dynamical systems for 4) isospectral bracket flows in infinite dimension, directly inspired by the QR method for spectral approximation of matrices, and 5) the computation of quasi-potential landscapes in cellular biology, for the multistability properties in the hematopoiesis mechanisms.