  We propose new linear combinations of compositions of a basic second-order
scheme with appropriately chosen coefficients to construct higher order
numerical integrators for differential equations. They can be considered as a
generalization of extrapolation methods and multi-product expansions. A general
analysis is provided and new methods up to order 8 are built and tested. The
new approach is shown to reduce the latency problem when implemented in a
parallel environment and leads to schemes that are significantly more efficient
than standard extrapolation when the linear combination is delayed by a number
of steps.
