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Problems with different time scales

Published online by Cambridge University Press:  07 November 2008

Heinz-Otto Kreiss
Heinz-Otto Kreiss
Affiliation:
Deparment of MathematicsUniversity of California at Los Angeles, Los Angeles, CA 90024USA, E-mail: kreiss@math.ucla.edu
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In this section we discuss a very simple problem. Consider the scalar initial value problem

Here ε > 0 is a small constant and a = a1 + ia2, a1, a2 real, is a complex number with |a| = 1. We can write down the solution of (1.1) explicity. It is

where

is the forced solution and

is a solution of the homogeneous equation

yS varies on the time scale ‘1’ while yF varies on the much faster scale 1/ε. We say that yS, yF vary on the slow and fast scale, respectively. We use also the phrase: yS and yF are the slow and the fast part of the solution, respectively.

Type
Research Article
Information
Acta Numerica , Volume 1 , January 1992 , pp. 101 - 139
Copyright
Copyright © Cambridge University Press 1992

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