论文标题
惯性Landau-Lifshitz-Gilbert方程的隐式有限差异方法的收敛分析
Convergence analysis of an implicit finite difference method for the inertial Landau-Lifshitz-Gilbert equation
论文作者
论文摘要
Landau-Lifshitz-Gilbert（LLG）方程是一种广泛使用的模型，用于铁磁材料中的快速磁化动力学。最近，已经提出了惯性LLG方程，其中包含惯性项，以捕获子picosecond TimeScale的超快速磁化动力学。从数学上讲，该广义模型包含第一个时间导数和新引入的磁化的第二个时间导数。因此，由于该方程的混合双曲线 - 寄生虫类型和退化性，它在数值分析中产生了额外的困难。在这项工作中，我们提出了一个基于时间和空间的核心差异的隐式有限差方案。应用固定点迭代方法来求解隐式非线性系统。借助二阶准确构造的解决方案，我们在$ \ ell^\ infty（0，t; h_h^1）$ norm中提供了$ h^1 $的收敛分析。结果表明，所提出的方法在时间和空间中都是准确的，具有无条件的稳定性和磁化长度的自然保留。在双曲线状态下，通过数值模拟观察到较短时间尺度上磁化的显着阻尼波行为。
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.
