The mimetic finite difference method for the Landau-Lifshitz equation

TL;DR

This paper introduces mimetic finite difference schemes for the Landau-Lifshitz equation, enabling flexible modeling of magnetic materials on complex meshes while preserving key physical properties.

Contribution

It develops and analyzes explicit and implicit mimetic finite difference schemes that work on general polytopal meshes and maintain the magnetization magnitude.

Findings

Schemes effectively model magnetization dynamics on distorted meshes.

The exchange energy decreases under certain conditions.

Numerical tests validate the schemes on standard and complex meshes.

Abstract

The Landau-Lifshitz equation describes the dynamics of the magnetization inside ferromagnetic materials. This equation is highly nonlinear and has a non-convex constraint (the magnitude of the magnetization is constant) which pose interesting challenges in developing numerical methods. We develop and analyze explicit and implicit mimetic finite difference schemes for this equation. These schemes work on general polytopal meshes which provide enormous flexibility to model magnetic devices with various shapes. A projection on the unit sphere is used to preserve the magnitude of the magnetization. We also provide a proof that shows the exchange energy is decreasing in certain conditions. The developed schemes are tested on general meshes that include distorted and randomized meshes. The numerical experiments include a test proposed by the National Institute of Standard and Technology and a test showing formation of domain wall structures in a thin film.