Adaptive Runge-Kutta methods for nonlinear two-point boundary value problems with mild boundary layers

Abstract

A variable mesh deferred correction algorithm based on implicit Runge-Kutta formulae is described for the approximate numerical solution of nonlinear two-point boundary value problems. A strategy for automatically choosing the variable mesh spacing is described and this seeks to equidistribute an approximation to the global truncation error of the Runge-Kutta formula. The facility of being able to use a variable mesh extends considerably the range of applicability of Runge-Kutta methods and in particular allows the possibility of solving rather more difficult problems, such as those with mild boundary layers or turning points, in an efficient manner. An extensive set of numerical results is given to illustrate the various algorithms described and a comparison is made with the deferred correction code of Lentini and Pereyra appearing in the NAG library.