Superconvergent Defect Correction Algorithms

In this paper we discuss several variants of the acceleration technique known as Iterated Defect Correction (IDeC) for the numerical solution of initial value problems for ODEs. A first approximation, computed by a low order basic method, is iteratively improved to obtain higher order solutions. We propose new versions of the IDeC algorithm with maximal achievable (super-)convergence order twice as high as in the classical setting. Moreover, if the basic numerical method is designed for a special type of ODE only, as it is the case for many geometric integrators, the idea of classical IDeC is not applicable in a straightforward way. Our approach enables the application of the defect correction principle in such cases as well. Key-Words: Iterated defect correction, splitting methods, geometric integration, superconvergent collocation.