Unconditional well-posedness and IMEX improvement of a family of predictor-corrector methods in micromagnetics

TL;DR

This paper proves the unconditional well-posedness of predictor-corrector methods for the Landau-Lifshitz-Gilbert equation in micromagnetics, extends their analysis, and introduces an IMEX strategy to improve computational efficiency.

Contribution

It provides a rigorous analysis of existing predictor-corrector methods, establishes their connection with other approaches, and develops an IMEX scheme to reduce computational costs.

Findings

Proved unconditional well-posedness of the methods.

Established a connection with other micromagnetics approaches.

Designed an IMEX strategy that reduces computational cost.

Abstract

Recently, Kim & Wilkening (Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation, Quart. Appl. Math., 76, 383-405, 2018) proposed two novel predictor-corrector methods for the Landau-Lifshitz-Gilbert equation (LLG) in micromagnetics, which models the dynamics of the magnetization in ferromagnetic materials. Both integrators are based on the so-called Landau-Lifshitz form of LLG, use mass-lumped variational formulations discretized by first-order finite elements, and only require the solution of linear systems, despite the nonlinearity of LLG. The first(-order in time) method combines a linear update with an explicit projection of an intermediate approximation onto the unit sphere in order to fulfill the LLG-inherent unit-length constraint at the discrete level. In the second(-order in time) integrator, the projection step is replaced by a linear constraint-preserving variational formulation. In this paper, we extend the analysis of the integrators by proving unconditional well-posedness and by establishing a close connection of the methods with other approaches available in the literature. Moreover, the new analysis also provides a well-posed integrator for the Schr\"odinger map equation (which is the limit case of LLG for vanishing damping). Finally, we design an implicit-explicit strategy for the treatment of the lower-order field contributions, which significantly reduces the computational cost of the schemes, while preserving their theoretical properties.