Explicit Magnus expansions for nonlinear equations

In this paper we develop and analyse new explicit Magnus expansions for the nonlinear equation $Y^\prime = A(t,Y) Y$ defined in a matrix Lie group. In particular, integration methods up to order four are presented in terms of integrals which can be either evaluated exactly or replaced by conveniently adapted quadrature rules. The structure of the algorithm allows to change the step size and even the order during the integration process, thus improving its efficiency. Several examples are considered, including isospectral flows and highly-oscillatory nonlinear differential equations.