# Second-order semi-implicit projection methods for micromagnetics simulations

## TL;DR

This paper introduces two second-order semi-implicit projection methods for micromagnetics simulations, improving accuracy and efficiency in solving the Landau-Lifshitz-Gilbert equation with complex constraints.

## Contribution

The paper develops and proves the unconditional solvability of two novel second-order semi-implicit methods based on BDF and interpolation formulas for micromagnetics.

## Key findings

- Methods are unconditionally uniquely solvable.
- Numerical examples confirm second-order accuracy.
- Methods outperform existing approaches in efficiency.

## Abstract

Micromagnetics simulations require accurate approximation of the magnetization dynamics described by the Landau-Lifshitz-Gilbert equation, which is nonlinear, nonlocal, and has a non-convex constraint, posing interesting challenges in developing numerical methods. In this paper, we propose two second-order semi-implicit projection methods based on the second-order backward differentiation formula and the second-order interpolation formula using the information at previous two temporal steps. Unconditional unique solvability of both methods is proved, with their second-order accuracy verified through numerical examples in both 1D and 3D. The efficiency of both methods is compared to that of another two popular methods. In addition, we test the robustness of both methods for the first benchmark problem with a ferromagnetic thin film material from National Institute of Standards and Technology.