The method of envelopes

Abstract:

The differential equation \[ \frac {{dx}}{{dt}} = \frac {A}{\varepsilon }x + g(t,x)\] where $A = \left [ {\begin {array}{*{20}{c}} 0 & { - 1} \\ 1 & 0 \\ \end {array} } \right ]$ and $\varepsilon > 0$ is a small parameter is a model for the stiff highly oscillatory problem. In this paper we discuss a new method for obtaining numerical approximations to the solution of the initial value problem for this differential equation. As $\varepsilon \to 0$, the asymptotic theory for this initial value problem yields an approximation to the solution which develops on two time scales, a fast time t and a slow time $\tau = t/\varepsilon$. We redevelop this asymptotic theory in such a form that the approximation consists of a series of simple functions of $\tau$, called carriers. (This series may be thought of as a Fourier series.) The coefficients of the terms of this series are functions of t. They are called envelopes and they modulate the carriers. Our computational method consists of determining numerical approximations to a finite collection of these envelopes. One of the principal merits of our method is its accuracy for the nonlinear problem.