一种基于快速傅里叶变换的直接分裂方法用于高效微磁模拟
Matrix-Free FFT-HSS Preconditioning for Periodic Landau-Lifshitz-Gilbert Saddle-Point Systems
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中文总结 AI 辅助
针对朗道 - 里夫希茨 - 吉尔伯特方程数值模拟难题，结合基于快速傅里叶变换的技术与分裂迭代法，为隐式欧拉格式开发无矩阵预条件器，研究克兰克 - 尼科尔森格式，经实验验证该方法能有效用于大规模微磁模拟。
中文摘要 AI 辅助
朗道 - 里夫希茨 - 吉尔伯特方程因其非线性、非凸单位长度约束和非局部场贡献给数值模拟带来重大挑战。现有隐式或半隐式格式虽无条件稳定，但求解非线性或线性化系统成本高。本文重新审视切平面公式，将离散问题写成广义鞍点系统。利用此结构，结合基于快速傅里叶变换的技术与分裂迭代法，为隐式欧拉格式开发无矩阵预条件器。为增强切向量几何一致性并减少投影步骤，进一步研究克兰克 - 尼科尔森格式，其算法结构与欧拉格式相同，可直接复用预条件器和求解器。通过大量二维和三维数值实验验证了该框架，表明其能准确保持约束、提高计算效率并具有稳健求解器性能，适用于复杂磁系统的大规模微磁模拟。
英文摘要
In this paper, a matrix-free Hermitian/skew-Hermitian splitting (HSS) preconditioner is proposed for periodic Landau--Lifshitz--Gilbert (LLG) saddle-point systems. The main contributions are threefold. (1) The coupled skew/constraint block has an explicit \(4\times4\) nodal inverse and requires no local factorization. Combining this local inverse with FFT inversion of the shifted exchange block gives \(O(N_g\log N_g)\) work and \(O(N_g)\) temporary storage per application. (2) We establish well-posedness and an even-step GMRES residual bound, with an iteration estimate uniform in the mesh size and time step for a class of coupled refinements. (3) The projected implicit Euler and projection-free Crank--Nicolson-type midpoint discretizations generate saddle-point systems of the same form, so the same matrix-free FFT--HSS preconditioning procedure applies to both. They achieve first- and second-order temporal accuracy, respectively; for quadratic-affine energies, the midpoint scheme also preserves nodal length and satisfies an exact discrete dissipation identity. Two- and three-dimensional experiments confirm the predicted temporal orders and midpoint invariants. On the largest smooth tests, FFT--HSS reduces GMRES iterations by \(35\)--\(39\%\) relative to same-grid Householder preconditioning and yields approximately 15-fold and 3-fold speedups over unpreconditioned GMRES in two and three dimensions, respectively. Broadband tests retain mesh-independent iterations in the covered refinement regime at time steps 13.3 times the linearized explicit exchange limit.
