惯性磁化动力学的中点几何积分器
Midpoint geometric integrators for inertial magnetization dynamics
AI总结：
针对惯性Landau-Lifshitz-Gilbert方程，提出两种基于隐式中点规则的无条件守恒二阶格式，并开发拟牛顿求解器，在宏自旋和薄膜模拟中验证了其高效性与准确性。
摘要：
我们考虑惯性版本的Landau-Lifshitz-Gilbert方程（iLLG）的数值求解，该方程描述了由于角动量弛豫而在磁化进动之上产生的高频章动。与经典LLG方程相比，iLLG方程定义了一个具有非常不同性质的高阶非线性动力系统，需要两倍的自由度用于时空离散化。它表现出基本的守恒性质，即磁化幅度保持、磁化投影守恒以及广义自由能的平衡方程，当外部磁场在时间上恒定时，这导致一个Lyapunov结构（即自由能是时间的递减函数）。我们提出了两种基于隐式中点规则的时间积分二阶数值格式来求解iLLG动力学。第一种格式无条件地保持所有守恒性质，使其成为模拟惯性磁化动力学的首选。然而，它意味着未知数加倍，需要对数值微磁代码进行重大修改，并增加计算成本，特别是对于空间非均匀动力学模拟。为了解决这个问题，我们提出了第二种时间步进方法，该方法在保持经典LLG动力学的隐式中点规则相同计算成本的同时，无条件地保持磁化幅度和投影。由于两种时间步进的隐式性质，我们开发了特殊的拟牛顿技术来求解每个时间步所需的非线性方程组。数值方案在宏自旋太赫兹频率响应的解析解上得到验证，第二种方案的有效性通过磁性薄膜中惯性自旋波传播的全微磁模拟得到证明。
英文摘要：
We consider the numerical solution of the inertial version of Landau-Lifshitz-Gilbert equation (iLLG), which describes high-frequency nutation on top of magnetization precession due to angular momentum relaxation. The iLLG equation defines a higher-order nonlinear dynamical system with very different nature compared to the classical LLG equation, requiring twice as many degrees of freedom for space-time discretization. It exhibits essential conservation properties, namely magnetization amplitude preservation, magnetization projection conservation, and a balance equation for generalized free energy, leading to a Lyapunov structure (i.e. the free energy is a decreasing function of time) when the external magnetic field is constant in time. We propose two second-order numerical schemes for integrating the iLLG dynamics over time, both based on implicit midpoint rule. The first scheme unconditionally preserves all the conservation properties, making it the preferred choice for simulating inertial magnetization dynamics. However, it implies doubling the number of unknowns, necessitating significant changes in numerical micromagnetic codes and increasing computational costs especially for spatially inhomogeneous dynamics simulations. To address this issue, we present a second time-stepping method that retains the same computational cost as the implicit midpoint rule for classical LLG dynamics while unconditionally preserving magnetization amplitude and projection. Special quasi-Newton techniques are developed for solving the nonlinear system of equations required at each time step due to the implicit nature of both time-steppings. The numerical schemes are validated on analytical solution for macrospin terahertz frequency response and the effectiveness of the second scheme is demonstrated with full micromagnetic simulation of inertial spin waves propagation in a magnetic thin-film.
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