Diagonally implicit runge-kutta formulae for the numerical integration of nonlinear two-point boundary value problems

Abstract

Fully implicit Runge-Kutta formulae, based on interpolatory quadrature schemes, for the approximate numerical solution of nonlinear two-point boundary value problems have been investigated by Weiss. Such formulae are not used very often in practice, however, because they generally require such a large computational effort as to make them uncompetitive with, for example, integration schemes based on the trapezoidal or implicit mid-point rules. In this paper we consider an alternative class of Runge-Kutta formulae, namely diagonally implicit Runge-Kutta (DIRK) formulae, which can be implemented more efficiently than the fully implicit formulae considered by Weiss. We also consider how these DIRK formulae can be implemented in a defect or deferred correction framework and we give some numerical results to illustrate the algorithms derived. One particular formula belonging to the DIRK class is the implicit mid-point rule. In this paper we derive an efficient implementation of this formula which is applicable when the given boundary conditions are non-separated.