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

Norbert J. Mauser, Carl-Martin Pfeiler, Dirk Praetorius, Michele Ruggeri

Research output: Contribution to journalArticlepeer-review

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 Schrödinger 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.
Original languageEnglish
Pages (from-to)33-54
Number of pages22
JournalApplied Numerical Mathematics
Volume180
Early online date25 May 2022
DOIs
Publication statusE-pub ahead of print - 25 May 2022

Keywords

  • Landau–Lifshitz–Gilbert equation
  • micromagnetics
  • Schrödinger map equation

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