SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1639-1662, January 2021.
The Landau--Lifshitz equation has been widely used to describe the dynamics of magnetization in a ferromagnetic material, which is highly nonlinear with the nonconvex constraint $|{m}|=1$. A crucial issue in designing efficient numerical schemes is to preserve this constraint in the discrete level. A simple and frequently used one is the projection method, which projects the numerical solution onto a unit sphere at each time step. The method has been used in many areas in the past several decades, while analysis has not been explored. In this paper, we present optimal error analysis of a backward Euler and a Crank--Nicolson semi-implicit projection finite difference scheme for the Landau--Lifshitz equation. The analysis is based on new and precise estimates of the difference between the errors of projected and unprojected solutions in both $L^2$ and $H^1$ norms. Some numerical experiments are provided to confirm our theoretical results.

中文翻译：

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Euler和Crank的最优误差分析--Landau--Lifshitz方程的Nicolson投影有限差分格式

SIAM 数值分析杂志，第 59 卷，第 3 期，第 1639-1662 页，2021 年 1 月。
Landau--Lifshitz 方程已被广泛用于描述铁磁材料中的磁化动力学，它具有非凸约束 $|{m}|=1$ 的高度非线性。设计有效数值方案的一个关键问题是在离散级别保留此约束。一种简单且常用的方法是投影法，它将数值解在每个时间步长投影到一个单位球面上。在过去的几十年中，该方法已被用于许多领域，而尚未对其进行分析。在本文中，我们提出了对 Landau--Lifshitz 方程的后向欧拉和 Crank--Nicolson 半隐式投影有限差分格式的最优误差分析。该分析基于对 $L^2$ 和 $H^1$ 范数中投影和非投影解的误差之间差异的新的精确估计。提供了一些数值实验来证实我们的理论结果。