Higher order numerical schemes have an advantage over low order schemes when the small scale behavior of solutions has to be captured. This paper addresses another advantage of higher order schemes, in connection with the issue of increasing modes.

Using truncation error arguments, the paper shows that the error in a numerical approximation depends not only on the simulated time and the order of the scheme, but also on the wavenumber. The dependence between accuracy and the Fourier spectrum of flows is discussed for several numerical schemes. In particular, the numerical solution of linear hyperbolic equations using a spatial discretization and a Runge-Kutta scheme is considered, and the dependence of truncation error, number of grid points, and wavenumber is studied. It is shown that changes of the wavenumber require a change of the number of grid points to ensure that the differentiation truncation error remains unchanged. For lower order schemes, the increase in the number of grid points is much higher than for higher order schemes.

This paper is written in an informal and intuitive style, with many examples as well as numerical experiments. Almost half of the paper is devoted to numerical experiments, for which Rayleigh-Taylor and Richtmyer-Meshkov calculations are used. People working in the area of numerical approximation of partial differential equations might benefit from the observed phenomena.