The mimetic finite difference method for the Landau-Lifshitz equation

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• 作者：Eugenia Kim^b ; ^a ; ^{kim107@math.berkeley.edu" class="auth_mail" title="E-mail the corresponding author} ; Konstantin Lipnikov^a
• 关键词：Micromagnetics ; Landau&ndash ; Lifshitz equation ; Landau&ndash ; Lifshitz&ndash ; Gilbert equation ; Mimetic finite difference method ; Polygonal meshes
• 刊名：Journal of Computational Physics
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：328
• 期：Complete
• 页码：109-130
• 全文大小：1380 K

文摘

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 poses 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.