2 - Problems with Different Time Scales

Abstract

This chapter discusses the problems with different time scales. The simplest problem with different time scales is given by the initial-value problem for the ordinary differential equation. In many applications, one is not interested in the fast time scale. Therefore, it is of interest to develop methods for preparing the initial data such that the fast time scale is suppressed. There are two ways to do this: initialization and filtering. The chapter also provides an overview on systems of ordinary differential equations. The effect of a highly oscillatory perturbation on the smooth solution is much smaller than the size of the perturbation. This means that in computations, one can be quite sloppy with the initialization and can apply a numerical filter after the calculation. Shallow water equations play a central role in geophysics. They govern two classes of motions with different time scales, consisting of low-frequency Rossby waves and high-frequency inertia gravity waves.