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. The transient solution decays rapidly and outside a boundary layer the solution depends essentially only on the slow scale. The chapter discusses a general theory that leads to a systematic way to prepare the initial data. There is an important difference between systems where the fast scale is represented by eigenvalues, which are real and negative or which are purely imaginary. In the first case, turning points are often present such as the eigenvalue change order of magnitude. There is an interior boundary layer where the solution changes rapidly. The main difficulty to solve these problems is to detect the boundary layers. Once one has passed the boundary layer, the fast scale is damped out. In the second case, the fast scale is not damped. The initial data is prepared such that the solution of the problem does not contain the fast time scale. These initial data are used for the difference approximation.