Efficient And Stable Third-order Method for Micromagnetics Simulations

TL;DR

The paper addresses the challenge of efficiently and stably solving the Landau-Lifshitz-Gilbert equation in the large-damping regime. It introduces a third-order time-stepping scheme built on a BDF3 backbone, with the damping term reformulated as a harmonic mapping flow and nonlinear terms treated explicitly, reducing linear solves to SPD Poisson problems solvable by FFTs. The method achieves 3rd-order temporal and 4th-order spatial accuracy while remaining unconditionally stable for large α, as validated by 1D and 3D tests and domain-wall motion studies that show linear velocity dependence on damping and external fields. This yields a highly efficient and robust tool for micromagnetic simulations in regimes where traditional lower-order methods struggle.

Abstract

To address the magnetization dynamics in ferromagnetic materials described by the Landau-Lifshitz-Gilbert equation under large damping parameters, a third-order accurate numerical scheme is developed by building upon a second-order method \cite{CaiChenWangXie2022} and leveraging its efficiency. This method boasts two key advantages: first, it only involves solving linear systems with constant coefficients, enabling the use of fast solvers and thus significantly enhancing numerical efficiency over existing first or second-order approaches. Second, it achieves third-order temporal accuracy and fourth-order spatial accuracy, while being unconditionally stable for large damping parameters. Numerical tests in 1D and 3D scenarios confirm both its third-order accuracy and efficiency gains. When large damping parameters are present, the method demonstrates unconditional stability and reproduces physically plausible structures. For domain wall dynamics simulations, it captures the linear relationship between wall velocity and both the damping parameter and external magnetic field, outperforming lower-order methods in this regard.

Figures (9)

• Figure 1: CPU time needed to achieve the desired numerical accuracy, for the proposed method, the BDF2 and the BDF1, 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: 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. 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 $\alpha$, which is physically reasonable. Top row: $\Delta t=1\;ps$; Bottom row: $\Delta t=0.1\;ps$.
• Figure 5: Energy evolution curves in terms of time, for the numerical results created by three numerical methods up to $t=2\,$ns with $k=1\;ps$ in the absence of external magnetic field for (a) $\alpha=5$, (b) $\alpha=8$, (c) $\alpha=10$, and (d) $\alpha=12$. In a relatively short period of time, as the order of the BDF method increases, the energy decays more slowly. As $\alpha$ increases, the BDF1 method reaches the lowest energy at equilibrium, followed by the BDF3 method, and the BDF2 method has the highest energy. At smaller $\alpha$ values, the energy curves of BDF3 and BDF2 are basically the same.

Theorems & Definitions (1)

• Remark 2.1