Instability of Numerical Method for Micromagnetics Simulations with Large Damping Parameters

TL;DR

This paper introduces a third-order accurate, unconditionally stable numerical scheme for the Landau-Lifshitz-Gilbert equation in micromagnetics, significantly improving efficiency for large damping parameters but facing limitations in capturing physical structures under certain conditions.

Contribution

A novel third-order accurate numerical method that solves constant-coefficient linear systems for micromagnetics simulations with large damping, enhancing efficiency and stability over existing methods.

Findings

Third-order accuracy and efficiency verified in 1D and 3D simulations.

Method fails to capture physical structures with large damping and pre-projection.

BDF1 outperforms BDF2 and BDF3 in domain wall dynamics simulations.

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.