Convergence analysis of an implicit finite difference method for the inertial Landau-Lifshitz-Gilbert equation

TL;DR

This paper introduces a second-order, unconditionally stable implicit finite difference scheme for the inertial Landau-Lifshitz-Gilbert equation, providing convergence analysis and capturing ultra-fast magnetization dynamics.

Contribution

It develops a novel implicit finite difference method with convergence proof for the inertial LLG equation, addressing its hyperbolic-parabolic complexity and preserving magnetization length.

Findings

Second order accuracy in time and space.

Unconditional stability of the scheme.

Observation of damping wave behaviors in simulations.

Abstract

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.