In the previous chapter we looked at the difficulties involved in the numerical solution to stationary transport equations dominated by convection. In the time-dependent case, we not only need to avoid oscillations in the numerical approximations, but we must also make sure that we can track the solution accurately as perturbations move over the computational mesh. We must make sure that the amplitude of the waves is not modified by the numerical scheme and that no phase lag is introduced. First, we will explain the concepts of numerical damping and phase lag in the numerical solutions and the techniques available to analyze them. Then we will extend the Petrov-Galerkin method to the time-dependent case, in such a way that those errors are minimized. We will discuss the main properties of the algorithms from both a theoretical and a practical point of view.