Discretizations with Dichotomic Stability for Two-Point Boundary Value Problems

Abstract

For a two-point boundary value problem to be well conditioned, the system of ordinary differential equations must necessarily possess a dichotomic set of fundamental solutions [7], with decaying modes controlled by initial conditions, and growing modes controlled by terminal conditions [10]. It was shown in [3] that it is important for a discretization of such a problem to preserve the dichotomy property, and the implications of this stability criterion were examined in a number of particular cases. Some simple difference schemes were examined for second order ordinary differential equations, and also various discretizations for first order systems, including those obtained by piecewise collocation and implicit Runge-Kutta type formulae. The last two examples were of multistep schemes, of such a form that they could be used in a sequential stepping mode, as would be done in shooting methods, or more generally in multiple shooting.