High order optimized geometric integrators for linear differential equations

In this paper new integration algorithms based on the Magnus expansion for linear differential equations up to eighth order are obtained. These methods are optimal with respect to the number of commutators required. Starting from Magnus series, integration schemes based on the Cayley transform and the Fer factorization are also built in terms of univariate integrals. The structure of the exact solution is retained while the computational cost is reduce compare to similar methods. Their relative performance is tested on some illustrative examples.