Stِrmer-Cowell: straight, summed and split. An overview

In this paper we consider the relationship between some (forms of) specific numerical methods for (second-order) initial value problems. In particular, the Stِrmer-Cowell method in second-sum form is shown to be the Gauss-Jackson method (and analogously, for the sake of completeness, we relate Adams-Bashforth-Moulton methods to their first-sum forms). Furthermore, we consider the split form of the Stِrmer-Cowell method. The reason for this consideration is the fact that these summed and split forms exhibit a better behaviour with respect to rounding errors than the original method (whether in difference or in ordinate notation). Numerical evidence will support the formal proofs that have been given elsewhere.