A novel third-order accurate and stable scheme for micromagnetic simulations

TL;DR

The paper addresses the need for accurate and stable time integration in micromagnetic simulations governed by the Landau-Lifshitz-Gilbert (LLG) equation across arbitrary damping. It introduces a novel third-order semi-implicit scheme based on a backward differentiation formula of order three (BDF3), with implicit fourth-order spatial discretization $\Delta_{h,(4)}$ and third-order extrapolated coefficients, complemented by a normalization step. Numerical experiments in 1D and 3D demonstrate strict third-order temporal accuracy and fourth-order spatial accuracy, with improved efficiency over first- and second-order methods and favorable comparison to a prior third-order scheme under moderate damping; energy dissipation patterns and domain-wall dynamics conform to theory, though the method shows sensitivity at very large $\alpha$. The results indicate that the proposed approach provides a reliable, efficient tool for large-scale, quantitative micromagnetic analyses, including accurate domain-wall velocity predictions, across practical damping regimes.

Abstract

High-fidelity numerical simulation serves as a cornerstone for exploring magnetization dynamics in micromagnetics. This work introduces a novel third-order temporally accurate and stable numerical scheme for the Landau-Lifshitz-Gilbert (LLG) equation, aiming to address the limitations in accuracy and efficiency often encountered with conventional approaches. Validation via nanostrip simulations confirms two principal advantages of the proposed method: it attains strict third-order temporal accuracy, surpassing many current techniques, and it offers superior computational efficiency, enabling rapid convergence without sacrificing numerical precision. For Gilbert damping coefficients $α$ ranging from $0.1$ to values below $10$, the scheme preserves strong stability and effectively avoids non-physical magnetization states. The magnetic microstructures predicted by this method are in excellent agreement with those from established benchmark methods, affirming its reliability for quantitative physical analysis. Salient distinctions between the proposed scheme and an existing third-order semi-implicit method include: (1) Solving the linear system associated with the existing scheme demands substantially greater computational time, underscoring the need for highly efficient solvers; (2) Although the proposed method shows increased sensitivity to damping parameters, it reliably converges to stable physical states and is effective in simulating magnetic domain wall motion, producing outcomes consistent with prior validated studies; (3) The energy levels computed by the proposed method are significantly lower than those obtained via the existing third-order scheme.

Figures (8)

• Figure 1: CPU time required to achieve the desired numerical accuracy for the proposed method, BDF2, and BDF1 in both 1D and 3D computations. CPU time is recorded as a function of approximation error by varying $k$ or $h$ independently. CPU time with varying $k$: proposed method $<$ BDF2 $<$ BDF1; CPU time with varying $h$: proposed method $<$ BDF1 $\lessapprox$ BDF2.
• Figure 2: Comparison of CPU time required to achieve the desired numerical accuracy for the proposed method for \ref{['eq-5']} and the third order semi-implicit scheme for \ref{['eq-model']}, along with their BDF2 and BDF1 counterparts. Top row: BDF3 method; Middle row: BDF2 method; Bottom row: BDF1 method. The observation is that the proposed method consumes more time than previous methods to achieve the same level of accuracy, indicating that its linear system is more difficult to solve. Numerical evidence reveals that GMRES solver convergence slows for larger temporal step size $k$ or smaller spatial grid-size $h$, increasing computational challenge. Thus, the proposed method requires a more efficient solver.
• Figure 3: Stable structures in the absence of magnetic field at $2\,$ns. Color denotes the angle between the first two components of the magnetization vector. Top two rows: Proposed method; Middle two rows: BDF2; Bottom two rows: BDF1. From left to right: $\alpha=0,0.01,0.1,1,5,10,40,100$. $k=1\;ps$.
• Figure 4: Stable structures in the absence of magnetic field at $2\,$ns. Color denotes the angle between the first two components of the magnetization vector. Top two rows: Proposed method; Middle two rows: BDF2; Bottom two rows: BDF1. From left to right: $\alpha=0,0.01,0.1,1,5,10,40,100$. $k=0.1\;ps$.
• Figure 5: Energy evolution curves of three numerical methods, with different damping constants, $\alpha=0.1,1,5,10$, up to $t=2\,$ns in the absence of external magnetic field. Left: Proposed numerical method; Middle: BDF1; Right: BDF2. One common feature is that the energy dissipation rate is faster for larger $\alpha$, which is physically reasonable. The proposed method is unstable with $\alpha=10$.

Theorems & Definitions (1)

• Remark 2.1