AI 中文总结
针对微磁模拟半隐式朗道-利夫希茨格式线性系统的收敛与性能问题，本文提出定制化聚合型多重网格方法，可减少迭代与计算成本，实现网格无关的鲁棒收敛。
AI 中文摘要
本研究开发了一种基于聚合的高效多重网格方法，用于求解微磁模拟中朗道-利夫希茨方程离散化产生的线性代数系统。该离散化将一阶向后微分公式与一阶外推法结合，所涉及的方程描述微磁模拟中的磁化动力学。对于这类线性系统，标准迭代方法和针对对称问题设计的传统多重网格求解器在网格细化时会出现收敛性下降、计算性能差的问题。现有算法难以处理系统的固有特性，即其稀疏谱特性和非对称矩阵结构。为克服这些局限，本文构建了一种多重网格框架，其中平滑算子和粗网格校正算子均针对目标线性系统定制。数值测试证实了所提方法的鲁棒性及与网格无关的收敛性。与成熟的Krylov子空间求解器和传统多重网格技术相比，该基于聚合的多重网格求解器大幅减少了迭代次数和整体计算成本，同时能准确呈现磁化动力学。
英文摘要
An efficient aggregation-based multigrid approach is developed in this work to solve linear algebraic systems generated by discretizing the Landau-Lifshitz equation in micromagnetics. The discretization couples the first-order backward differentiation formula with first-order extrapolation, and the underlying equation describes magnetization dynamics within micromagnetic simulations. For these linear systems, standard iterative methods and conventional multigrid solvers intended for symmetric problems suffer from deteriorated convergence under mesh refinement and poor computational performance. Existing algorithms struggle to handle the system's intrinsic features, namely its sparse spectral properties and non-symmetric matrix structure. To overcome these limitations, a multigrid framework is constructed, where smoothing and coarse-grid correction operators are customized for the target linear system. Numerical tests confirm the robustness and mesh-independent convergence of the resulting method. Compared with well-established Krylov-subspace solvers and conventional multigrid techniques, the aggregation-based multigrid solver cuts down iteration counts and overall computational cost considerably, while producing faithful representations of magnetization dynamics.
