The dynamics of magnetization in ferromagnetic materials are modeled by the Landau-Lifshitz equation, which presents significant challenges due to its inherent nonlinearity and non-convex constraint. These complexities necessitate efficient numerical methods for micromagnetics simulations. The Gauss-Seidel Projection Method (GSPM), first introduced in 2001, is among the most efficient techniques currently available. However, existing GSPMs are limited to first-order accuracy. This paper introduces two novel second-order accurate GSPMs based on a combination of the biharmonic equation and the second-order backward differentiation formula, achieving computational complexity comparable to that of solving the scalar biharmonic equation implicitly. The first proposed method achieves unconditional stability through Gauss-Seidel updates, while the second method exhibits conditional stability with a Courant-Friedrichs-Lewy constant of 0.25. Through consistency analysis and numerical experiments, we demonstrate the efficacy and reliability of these methods. Notably, the first method displays unconditional stability in micromagnetics simulations, even when the stray field is updated only once per time step.

翻译：铁磁材料中磁化强度的动力学行为由朗道-利夫希茨方程描述，该方程因其固有的非线性和非凸约束而带来显著的计算挑战。这些复杂性要求发展高效的数值方法以进行微磁学模拟。高斯-赛德尔投影方法（GSPM）于2001年首次提出，是目前最高效的数值技术之一。然而，现有的GSPM方法仅具有一阶精度。本文基于双调和方程与二阶后向差分公式的结合，提出了两种新型的二阶精度GSPM方法，其计算复杂度与隐式求解标量双调和方程相当。第一种方法通过高斯-赛德尔更新实现了无条件稳定性，而第二种方法则表现出条件稳定性，其柯朗-弗里德里希斯-列维常数为0.25。通过一致性分析和数值实验，我们验证了这些方法的有效性和可靠性。值得注意的是，即使在每个时间步仅更新一次杂散场的情况下，第一种方法在微磁学模拟中仍表现出无条件稳定性。