Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems

We present a family of numerical integrators based on the Magnus series expansions which is designed for solving non-autonomous differential equations. The main difference with standard Magnus integrators is that no commutators are involved. This property allows, in a simple way, to use the methods on non-linear ODEs. Fourth- and sixth-order methods for non-stiff differential equations are studied and new methods are presented. This type of method can easily be tailored to preserve geometric properties of the solutions, and we show through several examples that the performance and error behaviour is significantly better than known methods.