Superconvergent Deferred Correction Methods For First Order Systems of Nonlinear Two-Point Boundary Value Problems

Abstract

Iterated deferred correction is a widely used approach to the numerical solution of first order systems of nonlinear two-point boundary value problems. Normally the orders of accuracy of the various methods used in a deferred correction scheme differ by 2, and, as a direct result, each time a deferred correction is applied the order of the overall scheme is increased by a maximum of 2. In this paper we consider the construction of mono-implicit Runge--Kutta (MIRK) methods where an increase of four orders of accuracy is obtained for each deferred correction. We develop a very powerful yet rather straightforward theory which allows us to identify the appropriate Runge--Kutta formulae for inclusion in such schemes. In particular, we will focus on the construction of pairs of MIRK formulae of order 4 and 8 which will allow this superconvergence to be realized. We will further show that it is possible to derive formulae of this type for which high order interpolants and accurate error estimates are readily available.