The numerical approximation for the Landau-Lifshitz equation, which models the dynamics of the magnetization in a ferromagnetic material, is taken into consideration. This highly nonlinear equation, with a non-convex constraint, has several equivalent forms, and involves solving an auxiliary problem in the infinite domain. All these features have posed interesting challenges in developing numerical methods. In this paper, we first present a fully discrete semi-implicit method for solving the Landau-Lifshitz equation based on the second-order backward differentiation formula and the one-sided extrapolation (using previous time-step numerical values). A projection step is further used to preserve the length of the magnetization. Subsequently, we provide a rigorous convergence analysis for the fully discrete numerical solution by the introduction of two sets of approximated solutions where one set of solutions solves the Landau-Lifshitz equation and the other is projected onto the unit sphere. Second-order accuracy in both time and space is obtained provided that the spatial step-size is the same order as the temporal step-size. And also, the unique solvability of the numerical solution without any assumption for the step-size in both time and space is theoretically justified, using a monotonicity analysis. All these theoretical properties are verified by numerical examples in both 1D and 3D spaces.

考虑了 Landau-Lifshitz 方程的数值近似，该方程模拟了铁磁材料中的磁化动力学。这个具有非凸约束的高度非线性方程有几种等价形式，并涉及在无限域中求解辅助问题。所有这些特征都对开发数值方法提出了有趣的挑战。在本文中，我们首先提出了一种基于二阶向后微分公式和单边外推法（使用先前时间步长数值）来求解Landau-Lifshitz方程的完全离散的半隐式方法。投影步骤进一步用于保持磁化的长度。随后，我们通过引入两组近似解来为完全离散的数值解提供严格的收敛分析，其中一组解求解 Landau-Lifshitz 方程，另一组投影到单位球面上。如果空间步长与时间步长具有相同的阶数，则可以获得时间和空间的二阶精度。而且，使用单调性分析，在没有任何时间和空间步长假设的情况下，数值解的唯一可解性在理论上是合理的。所有这些理论特性都通过 1D 和 3D 空间中的数值例子进行了验证。如果空间步长与时间步长具有相同的阶数，则可以获得时间和空间的二阶精度。而且，使用单调性分析，在没有任何时间和空间步长假设的情况下，数值解的唯一可解性在理论上是合理的。所有这些理论特性都通过 1D 和 3D 空间中的数值例子进行了验证。如果空间步长与时间步长具有相同的阶数，则可以获得时间和空间的二阶精度。而且，使用单调性分析，在没有任何时间和空间步长假设的情况下，数值解的唯一可解性在理论上是合理的。所有这些理论特性都通过 1D 和 3D 空间中的数值例子进行了验证。