Enhancing Micromagnetics Simulations with a Third-Order Semi-Implicit Projection Method

TL;DR

This work addresses the need for high-accuracy, efficient micromagnetic simulations governed by the nondimensional LLG equation $\mathbf{m}_t = -\mathbf{m} \times \mathbf{h}_{\text{eff}} - \alpha \mathbf{m} \times (\mathbf{m} \times \mathbf{h}_{\text{eff}})$ by introducing a third-order temporal, fourth-order spatial semi-implicit projection method. The scheme derives a $\text{BDF3}$-type time discretization with a fourth-order spatial operator $\Delta_{h,(4)}$, and uses explicit extrapolation for nonlinear terms to preserve the unit-magnitude constraint. The authors compare against $\text{BDF1}$ and $\text{BDF2}$, showing $O(k^3+h^4)$ temporal and $O(h^4)$ spatial accuracy, with substantial gains in efficiency and stability for moderate-to-large damping $\alpha$ (validated over $\alpha$ in $[0.1,10]$ on 1D/3D tests and in domain-wall dynamics). They demonstrate that domain-wall velocity scales linearly with the external field $\mathbf{h}_e$ and quadratically with $\alpha$, and that the method captures stable micromagnetic microstructures consistent with established methods. The results suggest the method is well-suited for fast, reliable device-scale micromagnetics.

Abstract

Micromagnetics depends on high-fidelity numerical methods for magnetization dynamics. This work proposes a third-order temporal accuracy scheme for the Landau-Lifshitz-Gilbert equation, addressing accuracy-efficiency trade-offs in existing methods. Validated via nanostrip simulations (representative of real devices), the scheme offers two key advantages: rigorous third-order accuracy (surpassing existing simulation methods) and higher computational efficiency, ensuring fast convergence without precision loss. It maintains stability for Gilbert damping \(α\) from $0.1$ to $10$, avoiding non-physical states. The magnetic microstructures it captures are consistent with established methods, confirming reliability for physical analysis.

Figures (7)

• Figure 1: CPU time needed to achieve the desired numerical accuracy, for the proposed method, the BDF2 and the BDF1 method, in both the 1D and 3D computations. The CPU time is recorded as a function of the 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: Stable structures in the absence of magnetic field at $2\,$ns. The 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: $\alpha=0,0.01,0.1,1,5,10,40,100$. $dt=1\;ps$.
• Figure 3: Stable structures in the absence of magnetic field at $2\,$ns. The 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: $\alpha=0,0.01,0.1,1,5,10,40,100$. $dt=0.1\;ps$.
• Figure 4: 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.
• Figure 5: Energy evolution curves in terms of time, for the numerical results created by three numerical methods up to $t=2\,$ns in the absence of external magnetic field for (a) $\alpha=0.1$, (b) $\alpha=1$, (c) $\alpha=5$, and (d) $\alpha=10$. The energy dissipation pattern of the proposed method is consistent with the BDF2 method for $\alpha=0.1,1,5$, and inconsistent with other two methods for $\alpha=10$. The rate of energy dissipation of the BDF1 method is slower than that of other methods. The energy of BDF2 and the proposed method when they reach a steady state is lower than that of BDF1 when it reaches a steady state.