A Numerical Method for Singular Perturbation Problems with Turning Points

Abstract

Until recently, the development of general numerical methods for singular perturbation problems whose solutions exhibit internal layer type behavior has been largely neglected. Indeed, even the analytic study of general systems of first order ODEs with this type of behavior appears to be quite limited; perhaps this is one of the reasons for the lack of progress in this area. In this paper, we report on results obtained with H. O. Kreiss and N. Nichols [7], which address this problem. We will give a presentation which is somewhat different than that in [7], with the hope of emphasizing the similarity of our approach to the ideas that underlie the analytic technique of matched asymptotic expansions. We also present some recent results for nonlinear problems with internal layer behavior which have been obtained together with W. L. Kath and H. O. Kreiss. All of our results are for the two-point boundary value problem for systems of first-order ordinary differential equations. Since the solutions of singular perturbation problems of this type typically vary on two or more scales, these problems are often called “stiff” boundary value problems as well.