In this thesis we highlight the necessity of employing numerical schemes that preserve the large-time dynamical properties of the continuous system. We focus on Burgers- like equations, which are well known to develop N-waves as intermediate or asymptotic profiles. As we show, not only forward simulations can be distorted; the efficiency of solutions of optimization and inverse design problems might be affected too. In particular, we apply our results to the case of the prediction and control of the sonic- boom phenomenon, modeled by the augmented Burgers equation
