Efficient And Stable Third-order Method for Micromagnetics Simulations

TL;DR

This paper introduces a third-order accurate, unconditionally stable numerical scheme for micromagnetics simulations that is more efficient and accurate than existing methods, especially for large damping parameters.

Contribution

It develops a third-order method with constant coefficient linear systems, achieving high accuracy and stability, improving computational efficiency in micromagnetics simulations.

Findings

Third-order temporal and fourth-order spatial accuracy achieved.

Method is unconditionally stable for large damping parameters.

Outperforms lower-order methods in domain wall velocity simulations.

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.