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
Moreover, if the basic numerical method is designed for a special type
of ODEs only, as 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.