Runge-Kutta methods for the solution of stiff two-point boundary value problems

Abstract

One of the most popular approaches to the numerical solution of two-point boundary value problems is shooting. However this approach is often ineffective for singularly perturbed problems due to the possible presence of rapidly increasing modes which cannot be dealt with using an initial value solver. In this paper we survey the use of implicit Runge-Kutta methods for such problems. It is shown that certain classes of implicit Runge-Kutta formulae are both stable and accurate for singularly perturbed problems and an efficient implementation of these formulae, based on the use of deferred correction, is described.