Magnetization dynamics in ferromagnetic materials is modeled by the Landau-Lifshitz (LL) equation, a nonlinear system of partial differential equations. Among the numerical approaches, semi-implicit schemes are widely used in the micromagnetics simulation, due to a nice compromise between accuracy and efficiency. At each time step, only a linear system needs to be solved and a projection is then applied to preserve the length of magnetization. However, this linear system contains variable coefficients and a non-symmetric structure, and thus an efficient linear solver is highly desired. If the damping parameter becomes large, it has been realized that efficient solvers are only available to a linear system with constant, symmetric, and positive definite (SPD) structure. In this work, based on the implicit-explicit Runge-Kutta (IMEX-RK) time discretization, we introduce an artificial damping term, which is treated implicitly. The remaining terms are treated explicitly. This strategy leads to a semi-implicit scheme with the following properties: (1) only a few linear system with constant and SPD structure needs to be solved at each time step; (2) it works for the LL equation with arbitrary damping parameter; (3) high-order accuracy can be obtained with high-order IMEX-RK time discretization. Numerically, second-order and third-order IMEX-RK methods are designed in both the 1-D and 3-D domains. A comparison with the backward differentiation formula scheme is undertaken, in terms of accuracy and efficiency. The robustness of both numerical methods is tested on the first benchmark problem from National Institute of Standards and Technology. The linearized stability estimate and optimal rate convergence analysis are provided for an alternate IMEX-RK2 numerical scheme as well.

翻译：铁磁材料中的磁化动力学由Landau-Lifshitz (LL)方程建模，该方程是非线性偏微分方程组。在数值方法中，半隐式格式因其在精度与效率间的良好折衷而被广泛应用于微磁学模拟。每个时间步仅需求解一个线性系统，随后施加投影以保持磁化长度。然而，该线性系统包含变系数和非对称结构，因此高效线性求解器的需求十分迫切。当阻尼参数较大时，人们意识到高效求解器仅适用于具有常数、对称正定（SPD）结构的线性系统。本文基于隐式-显式龙格-库塔（IMEX-RK）时间离散方法，引入一个隐式处理的人工阻尼项，其余项则显式处理。该策略导出的半隐式格式具有以下性质：(1) 每个时间步仅需求解少量具有常数和SPD结构的线性系统；(2) 适用于任意阻尼参数的LL方程；(3) 通过高阶IMEX-RK时间离散可实现高阶精度。数值上，我们在一维和三维区域设计了二阶和三阶IMEX-RK方法。在精度和效率方面，与向后差分公式格式进行了比较。两种数值方法的鲁棒性均通过美国国家标准与技术研究院的第一个基准测试问题验证。此外，还提供了另一种IMEX-RK2数值格式的线性化稳定性估计和最优收敛率分析。