This work proposes and analyzes a fully discrete numerical scheme for solving the Landau-Lifshitz-Gilbert (LLG) equation, which achieves fourth-order spatial accuracy and third-order temporal accuracy.Spatially, fourth-order accuracy is attained through the adoption of a long-stencil finite difference method, while boundary extrapolation is executed by leveraging a higher-order Taylor expansion to ensure consistency at domain boundaries. Temporally, the scheme is constructed based on the third-order backward differentiation formula (BDF3), with implicit discretization applied to the linear diffusion term for numerical stability and explicit extrapolation employed for nonlinear terms to balance computational efficiency. Notably, this numerical method inherently preserves the normalization constraint of the LLG equation, a key physical property of the system.Theoretical analysis confirms that the proposed scheme exhibits optimal convergence rates under the \(\ell^{\infty}([0,T],\ell^2)\) and \(\ell^2([0,T],H_h^1)\) norms. Finally, numerical experiments are conducted to validate the correctness of the theoretical convergence results, demonstrating good agreement between numerical observations and analytical conclusions.

翻译：本文提出并分析了一种用于求解Landau-Lifshitz-Gilbert（LLG）方程的全离散数值格式，该格式实现了四阶空间精度与三阶时间精度。在空间维度上，通过采用长模板有限差分法达到四阶精度，并利用高阶泰勒展开进行边界外推以确保计算域边界的一致性。在时间维度上，该格式基于三阶后向差分公式（BDF3）构建，其中线性扩散项采用隐式离散以保证数值稳定性，非线性项则采用显式外推以平衡计算效率。值得注意的是，该数值方法天然保持了LLG方程的归一化约束条件，这是该系统的关键物理特性。理论分析证实，所提格式在\(\ell^{\infty}([0,T],\ell^2)\)和\(\ell^2([0,T],H_h^1)\)范数下均表现出最优收敛阶。最后，通过数值实验验证了理论收敛结果的正确性，表明数值观测结果与理论分析结论高度吻合。