  Recent theoretical and experimental advances show that the inertia of
magnetization emerges at sub-picoseconds and contributes to the ultrafast
magnetization dynamics which cannot be captured intrinsically by the LLG
equation. Therefore, as a generalization, the inertial Landau-Lifshitz-Gilbert
(iLLG) equation is proposed to model the ultrafast magnetization dynamics.
Mathematically, the LLG equation is a nonlinear system of parabolic type with
(possible) degeneracy. However, the iLLG equation is a nonlinear system of
mixed hyperbolic-parabolic type with degeneracy, and exhibits more complicated
structures. It behaves like a hyperbolic system at the sub-picosecond scale
while behaves like a parabolic system at larger timescales. Such hybrid
behaviors impose additional difficulties on designing numerical methods for the
iLLG equation. In this work, we propose a second-order semi-implicit scheme to
solve the iLLG equation. The second temporal derivative of magnetization is
approximated by the standard centered difference scheme and the first
derivative is approximated by the midpoint scheme involving three time steps.
The nonlinear terms are treated semi-implicitly using one-sided interpolation
with the second-order accuracy. At each step, the unconditionally unique
solvability of the unsymmetric linear system of equations in the proposed
method is proved with a detailed discussion on the condition number.
Numerically, the second-order accuracy in both time and space is verified.
Using the proposed method, the inertial effect of ferromagnetics is observed in
micromagnetics simulations at small timescales, in consistency with the
hyperbolic property of the model at sub-picoseconds. For long time simulations,
the results of the iLLG model are in nice agreements with those of the LLG
model, in consistency with the parabolic feature of the iLLG model at larger
timescales.
