Instability of Numerical Method for Micromagnetics Simulations with Large Damping Parameters

TL;DR

The paper addresses stable, high-order numerical simulation of the non-dimensional Landau-Lifshitz-Gilbert equation under large damping ($\alpha$). It introduces a third-order backward differentiation formula (BDF3)–based scheme that treats the diffusion part with a constant-coefficient Laplacian via BDF3 while handling nonlinear terms explicitly with a third-order extrapolation, resulting in only FFT-based Poisson solves per step and achieving $O(k^3 + h^4)$ accuracy. Numerical experiments in 1D and 3D validate the method’s third-order temporal and fourth-order spatial accuracy and show favorable efficiency versus BDF2 and BDF1, though stability degrades for large $\alpha$ in pre-projected or domain-wall settings. The study also highlights that pre-projected nonlinear steps can induce instability, and that simpler, lower-order schemes like BDF1 may offer more robust stability in extreme damping regimes. Overall, the method provides a fast, accurate tool for micromagnetics at large damping, with caveats that motivate further stabilization work.

Abstract

We propose and implement a third-order accurate numerical scheme for the Landau-Lifshitz-Gilbert equation, which describes magnetization dynamics in ferromagnetic materials under large damping parameters. This method offers two key advantages: (1) It solves only constant-coefficient linear systems, enabling fast solvers and thus achieving much higher numerical efficiency than existing second-order methods. (2) It attains third-order temporal accuracy and fourth-order spatial accuracy, and is unconditionally stable for large damping parameters. Numerical examples in 1D and 3D simulations verify both its third-order accuracy and efficiency gains. However, when large damping parameters and pre-projection solutions are involved, both this proposed method and a second-order method of the same style fail to capture reasonable physical structures, despite extensive theoretical analyses. Additionally, comparisons of domain wall dynamics among BDF2, BDF3, and BDF1 show that BDF2 and BDF3 yield failed simulations, while BDF1 performs marginally better.

Figures (9)

• 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 when $\alpha=1,5,10,40,100$. The color denotes the angle between the first two components of the magnetization vector. Top: Proposed method; Middle: BDF2; Bottom: BDF1; From left to right: $\alpha=1,5,10,40,100$, $dt=1\;ps$.
• Figure 3: Stable structures in the absence of magnetic field at $2\,$ns when $\alpha=1,5,10,40,100$. The color denotes the angle between the first two components of the magnetization vector. Top: Proposed method; Middle: BDF2; Bottom: BDF1; From left to right: $\alpha=1,5,10,40,100$, $dt=0.1\;ps$.
• Figure 4: Energy evolution curves of three numerical methods, with different damping constants, $\alpha=5,8,10,12$, up to $t=2\,$ns in the absence of external magnetic field. Left: Proposed numerical method; Middle: BDF2; Right: BDF1. One common feature is that the energy dissipation rate is faster for larger $\alpha$. Top row: $dt=1\;ps$; Bottom row: $dt=0.1\;ps$
• Figure 5: Energy evolution curves in terms of time, for the numerical results created by three numerical methods up to $t=2\,$ns with $dt=1\;ps$ in the absence of external magnetic field for (a) $\alpha=5$, (b) $\alpha=8$, (c) $\alpha=10$, and (d) $\alpha=12$. The energy dissipation pattern of the proposed method is different with the other two methods, and the BDF1 has a relatively reasonable energy dissipation pattern from the other two methods. As $\alpha$ increases, the stability of the method becomes worse. As $\alpha$ increases, the energy value of the proposed method at the final moment decreases.

Theorems & Definitions (1)

• Remark 2.1