Resumen
We investigate finite difference schemes which approximate $2\times 2$ one-dimensional linear dissipative hyperbolic systems. We show that it is possible to introduce some suitable modifications in standard upwinding schemes, which keep into account the long-time behavior of the solutions, to yield numerical approximations which are increasingly accurate for large times when computing small perturbations of stable asymptotic states, respectively, around stationary solutions and in the diffusion (Chapman–Enskog) limit.