Methods are developed for increasing the fidelity of difference approximations to hyperbolic partial differential equations. A relation between the truncation error and the exact and approximate amplification factors is derived. Based upon this relation, quantitative criteria for the minimization of dissipation and dispersion are derived, and difference schemes which satisfy these criteria are constructed. Completely new schemes, one of them promising, are obtained, together with several well-known schemes. One of these is the Fromm scheme, for which previously only a heuristic derivation could be given. It is shown that in general the accuracy of the Rusanov-Burstein-Mirin scheme is disappointing. A simple modification was found to remedy this deficiency.