Improving the Performance of Numerical Methods for Two Point Boundary Value Problems

We report on an ongoing investigation into the performance of numerical methods for two point boundary value problems. We outline how methods based on multiple shooting, collocation and other local discretizations can share a common structure. The identification of this common structure permits us to analyse how various components of a method interact and also permits us to consider the assembly of a collection of modular routines which will eventually form the basis for a software environment for solving two point boundary value problems.The initial stage of our investigation has involved the implementation and analysis of a family of multiple shooting methods as well as a family of collocation/Runge-Kutta methods. We have analysed the performance of these methods on a class of singular perturbation problems. The numerical conditioning of both families of methods on such problems and the convergence requirements of the corresponding iteration schemes (used to solve the discretized problem) has been investigated. Appropriate modifications to these methods which permit the effective solution of such problems will be discussed. We will also identify a subfamily of the Runge-Kutta methods that are particularly effective.