Composition methods for differential equations with processing

We construct numerical integrators for differential equations up to order twelve obtained by composition of basic integrators. The following cases are considered: (i) composition for a system separable in two solvable parts, (ii) composition using as basic methods a first order integrator and its adjoint, (iii) composition using second order symmetric methods and (iv) composition using fourth order symmetric methods. Each scheme is implemented with a processor or corrector to improve their efficiency, and this can be done at virtually cost free.