  We discuss a mass-lumped midpoint scheme for the numerical approximation of
the Landau-Lifshitz-Gilbert equation, which models the dynamics of the
magnetization in ferromagnetic materials. In addition to the classical
micromagnetic field contributions, our setting covers the non-standard
Dzyaloshinskii-Moriya interaction, which is the essential ingredient for the
enucleation and stabilization of magnetic skyrmions. Our analysis also includes
the inexact solution of the arising nonlinear systems, for which we discuss
both a constraint preserving fixed-point solver from the literature and a novel
approach based on the Newton method. We numerically compare the two
linearization techniques and show that the Newton solver leads to a
considerably lower number of nonlinear iterations. Moreover, in a numerical
study on magnetic skyrmions, we demonstrate that, for magnetization dynamics
that are very sensitive to energy perturbations, the midpoint scheme, due to
its conservation properties, is superior to the dissipative tangent plane
schemes from the literature.
