  Recently, Kim & Wilkening (Convergence of a mass-lumped finite element method
for the Landau-Lifshitz equation, Quart. Appl. Math., 76, 383-405, 2018)
proposed two novel predictor-corrector methods for the Landau-Lifshitz-Gilbert
equation (LLG) in micromagnetics, which models the dynamics of the
magnetization in ferromagnetic materials. Both integrators are based on the
so-called Landau-Lifshitz form of LLG, use mass-lumped variational formulations
discretized by first-order finite elements, and only require the solution of
linear systems, despite the nonlinearity of LLG. The first(-order in time)
method combines a linear update with an explicit projection of an intermediate
approximation onto the unit sphere in order to fulfill the LLG-inherent
unit-length constraint at the discrete level. In the second(-order in time)
integrator, the projection step is replaced by a linear constraint-preserving
variational formulation. In this paper, we extend the analysis of the
integrators by proving unconditional well-posedness and by establishing a close
connection of the methods with other approaches available in the literature.
Moreover, the new analysis also provides a well-posed integrator for the
Schr\"odinger map equation (which is the limit case of LLG for vanishing
damping). Finally, we design an implicit-explicit strategy for the treatment of
the lower-order field contributions, which significantly reduces the
computational cost of the schemes, while preserving their theoretical
properties.
