On solving boundary value problems for multi-scale systems using asymptotic approximations and multiple-shooting

This paper describes a method for obtaining the numerical solution of certain singularly perturbed boundary value problems for linear systems of ordinary differential equations whose solutions involve dynamics with multiple time scales. A technique to decouple different time scales using a Riccati transformation is presented. For the slow modes, which provide smooth solutions, regular perturbation expansion and multiple shooting strategies are combined. Fast modes, which are assumed to be significant only in endpoint layers, are computed after appropriate stretchings have been made in the system. Advantages of this approach include adaptability, flexible use of output points, and automatic determination of layer thicknesses. Dedicated to Professor Germund Dahlquist, with the hope that he will long continue to make significant and singular perturbations to the numerical analysis of differential equations.