A highly stable deferred correction scheme with interpolant for systems of nonlinear two-point boundary value problems

摘要

Iterated deferred correction is a very popular approach to the numerical solution of general first-order systems of nonlinear two-point boundary value problems. Typically the order of accuracy of the solution is increased by 2 each time that a deferred correction is applied and this allows inexpensive local error estimates to be computed using embedding. Recent research has shown how deferred correction schemes having the very desirable property of superconvergence can be derived. This marks a significant step forward in the design of deferred correction algorithms although the price to be paid for superconvergence is that it is difficult to derive realistic embedded local error estimates. In the present paper we show that Richardson extrapolation is a viable alternative for error estimation and we consider in some detail the relative merits of embedding and extrapolation. Guided by this analysis we are able to derive for the first time a Lobatto deferred correction code which is very efficient for the solution of stiff problems, particularly in cases where the MIRK code TWPBVP.f is unstable. Furthermore we are able to derive efficient interpolants for this Lobatto code which are also applicable to MIRK formulae and we consider the problem of estimating the error in these interpolants. This completely solves the interpolation problem for our deferred correction codes apart perhaps for deriving interpolants for extremely stiff problems. This new Lobatto code is available on the web page of one of the present authors and fills a large gap in currently available deferred correction software.