Collocation and Relaxed Collocation for the Fer and the Magnus Expansions

We consider the Fer and the Magnus expansions for the numerical solution of the nonlinear matrix Lie-group ODE $y' = \gamma(t,y) y$, $y(0)= y_0,$ where y evolves in a matrix Lie group G and $\gamma(t,y)$ is in the Lie algebra %$\hbox{\Fr{g}}$. $\ghe$. Departing from a geometrical approach that distinguishes between those operations performed in the group and those performed in the tangent space, we construct Lie-group invariant methods based on collocation. We prove that, as long as the two expansions are correctly truncated, the collocation nodes $c_1, c_2, \ldots, c_{\nu}$ yield numerical methods whose order is the same as in the classical setting. We also relax the collocation conditions, thereby devising explicit methods of order 3. To conclude, we discuss the proposed methods in a numerical experiment that arises in Hamiltonian mechanics, comparing the results with projection techniques.