  We propose a geometric integrator to numerically approximate the flow of Lie
systems. The key is a novel procedure that integrates the Lie system on a Lie
group intrinsically associated with a Lie system on a general manifold via a
Lie group action, and then generates the discrete solution of the Lie system on
the manifold via a solution of the Lie system on the Lie group. One major
result from the integration of a Lie system on a Lie group is that one is able
to solve all associated Lie systems on manifolds at the same time, and that Lie
systems on Lie groups can be described through first-order systems of linear
homogeneous ordinary differential equations (ODEs) in normal form. This brings
a lot of advantages, since solving a linear system of ODEs involves less
numerical cost. Specifically, we use two families of numerical schemes on the
Lie group, which are designed to preserve its geometrical structure: the first
one based on the Magnus expansion, whereas the second is based on
Runge-Kutta-Munthe-Kaas (RKMK) methods. Moreover, since the aforementioned
action relates the Lie group and the manifold where the Lie system evolves, the
resulting integrator preserves any geometric structure of the latter. We
compare both methods for Lie systems with geometric invariants, particularly a
class on Lie systems on curved spaces. We also illustrate the superiority of
our method for describing long-term behavior and for differential equations
admitting solutions whose geometric features depends heavily on initial
conditions. As already mentioned, our milestone is to show that the method we
propose preserves all the geometric invariants very faithfully, in comparison
with nongeometric numerical methods.
