  The Landau-Lifshitz-Gilbert (LLG) equation is a widely used model for fast
magnetization dynamics in ferromagnetic materials. Recently, the inertial LLG
equation, which contains an inertial term, has been proposed to capture the
ultra-fast magnetization dynamics at the sub-picosecond timescale.
Mathematically, this generalized model contains the first temporal derivative
and a newly introduced second temporal derivative of magnetization.
Consequently, it produces extra difficulties in numerical analysis due to the
mixed hyperbolic-parabolic type of this equation with degeneracy. In this work,
we propose an implicit finite difference scheme based on the central difference
in both time and space. A fixed point iteration method is applied to solve the
implicit nonlinear system. With the help of a second order accurate constructed
solution, we provide a convergence analysis in $H^1$ for this numerical scheme,
in the $\ell^\infty (0, T; H_h^1)$ norm. It is shown that the proposed method
is second order accurate in both time and space, with unconditional stability
and a natural preservation of the magnetization length. In the hyperbolic
regime, significant damping wave behaviors of magnetization at a shorter
timescale are observed through numerical simulations.
