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.
翻译:Landau-Lifshitz- Gilbert (LLG) 方程式是铁磁材料快速磁化动力的一种广泛使用的模型。 最近, 惯性LLG 方程式( 包含惯性术语) 提议在亚对地第二个时间尺度上捕捉超快磁化动力。 从数学角度讲, 这一通用模型包含第一个时间衍生物, 以及新引入的第二个时间衍生物。 因此, 由于该方程式的双双曲- 单曲类型与分解性混合, 它在数字分析中造成了额外的困难。 在这项工作中, 我们提议基于时间和空间中心差异的隐含有限差异方案。 一种固定点的代用法用于解决隐含的非线性系统。 在第二个精确的构造解决方案的帮助下, 我们用美元/ ellinfty ( 0, T; H_h ⁇ 1) 标准, 为该数字公式提供了以1美元表示的趋同值分析。 因此, 拟议的方法在时间和空间上都是第二顺序的精确度, 并且以无条件的稳定性和在磁磁化的自然保存时间缩缩缩度中, 。