  Micromagnetics simulations require accurate approximation of the
magnetization dynamics described by the Landau-Lifshitz-Gilbert equation, which
is nonlinear, nonlocal, and has a non-convex constraint, posing interesting
challenges in developing numerical methods. In this paper, we propose two
second-order semi-implicit projection methods based on the second-order
backward differentiation formula and the second-order interpolation formula
using the information at previous two temporal steps. Unconditional unique
solvability of both methods is proved, with their second-order accuracy
verified through numerical examples in both 1D and 3D. The efficiency of both
methods is compared to that of another two popular methods. In addition, we
test the robustness of both methods for the first benchmark problem with a
ferromagnetic thin film material from National Institute of Standards and
Technology.
