Convergence Analysis of an Implicit Finite Difference Method for the Inertial Landau–Lifshitz–Gilbert Equation

The Landau–Lifshitz–Gilbert (LLG) equation is widely used to model the fast magnetization dynamics of ferromagnets. Recent experimental observations have revealed ultra-fast dynamics at the sub-picosecond timescale, and the inertial LLG equation is proposed to capture the ultra-fast behaviour of magnetization, in which a second temporal derivative of magnetization (inertial term) is introduced. The inertial LLG equation is therefore a mixed hyperbolic-parabolic type equation with degeneracy, which produces extra difficulties in numerical analysis. In this paper, we propose an implicit finite difference scheme based on the central difference in both time and space, and a fixed point iteration method to solve the nonlinear system. By a constructed solution with second order accuracy, we get a linear system and provide an unconditional convergence analysis in \(\ell ^\infty ([0, T]; H_h^1(\varOmega ))\) ${\ell }^{\infty }([0,T];H_h^1(\Omega ))$ .. We demonstrate that the proposed method is second order accurate in both time and space, a natural preservation of the magnetization length and the energy decaying. In the hyperbolic regime, significant nutation of magnetization at a shorter timescale are simulated by numerical simulations.