Conditioning and Hybrid Mesh Selection Algorithms for Two-Point Boundary Value Problems

Abstract

Boundary value problems for ordinary differential equations (BVODES) occur in
a great many practical situations and they are generally much harder to solve
than initial value problems. Traditionally codes for BVODES
did not take into account the conditioning of the problem and it was
generally assumed that the problem being solved was well conditioned so that
small local errors gave rise to correspondingly small global errors.
Recently a new generation of codes which take account of conditioning has
been developed. However most of these codes are based on a rather ad hoc
approach with the need to choose several heuristics without any real guidance
on how these choices can be made. In this paper we identify clearly which
heuristics need to be chosen and we discuss different choices of monitor
functions that are used in our codes. This has the important effect of
unifying the various approaches that have recently been proposed. This in
turn allows us, in the present paper, to introduce a new technique for
computing the conditioning which is ideally suited to BVODES.