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We present a steepest descent energy minimization scheme for micromagnetics. The method searches on a curve that lies on the sphere which keeps the magnitude of the magnetization vector constant. The step size is selected according to a…

Computational Physics · Physics 2014-02-04 Lukas Exl , Simon Bance , Franz Reichel , Thomas Schrefl , Hans Peter Stimming , Norbert J. Mauser

We consider the coupled system of the Landau--Lifshitz--Gilbert equation and the conservation of linear momentum law to describe magnetic processes in ferromagnetic materials including magnetoelastic effects in the small-strain regime. For…

Numerical Analysis · Mathematics 2025-01-15 Hywel Normington , Michele Ruggeri

Speeding up computationally expensive problems, such as numerical simulations of large micromagnetic systems, requires efficient use of parallel computing infrastructures. While parallelism across space is commonly exploited in…

Computational Physics · Physics 2023-10-19 Robert Kraft , Sabri Koraltan , Markus Gattringer , Florian Bruckner , Dieter Suess , Claas Abert

We consider a lowest-order finite element discretization of the nonlinear system of Maxwell's and Landau-Lifshitz-Gilbert equations (MLLG). Two algorithms are proposed to numerically solve this problem, both of which only require the…

Numerical Analysis · Mathematics 2017-01-30 L'ubomir Banas , Marcus Page , Dirk Praetorius

Micromagnetic simulations based on the stochastic Landau-Lifshitz-Gilbert equation are used to calculate dynamic magnetic hysteresis loops relevant to magnetic hyperthermia. With the goal to effectively simulate room-temperature loops for…

Materials Science · Physics 2021-11-09 Razyeh Behbahani , Martin L. Plumer , Ivan Saika-Voivod

A second order accurate numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with large damping parameters. The main advantages of this method…

Computational Physics · Physics 2022-01-26 Yongyong Cai , Jingrun Chen , Cheng Wang , Changjian Xie

The Landau-Lifshitz-Gilbert (LLG) equation is a widely used model for fast magnetization dynamics in ferromagnetic materials. Recently, the inertial LLG equation, which contains an inertial term, has been proposed to capture the ultra-fast…

Numerical Analysis · Mathematics 2022-09-13 Jingrun Chen , Panchi Li , Cheng Wang

To describe and simulate dynamic micromagnetic phenomena, we consider a coupled system of the nonlinear Landau-Lifshitz-Gilbert equation and the conservation of momentum equation. This coupling allows to include magnetostrictive effects…

Numerical Analysis · Mathematics 2014-12-10 L'ubomir Banas , Marcus Page , Dirk Praetorius , Jonathan Rochat

A third-order accurate implicit-explicit Runge-Kutta time marching numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with arbitrary damping…

Numerical Analysis · Mathematics 2024-07-09 Yan Gui , Rui Du , Cheng Wang

To address the magnetization dynamics in ferromagnetic materials described by the Landau-Lifshitz-Gilbert equation under large damping parameters, a third-order accurate numerical scheme is developed by building upon a second-order method…

Numerical Analysis · Mathematics 2025-10-31 Changjian Xie , Cheng Wang

The Landau-Lifshitz equation describes the dynamics of magnetization in ferromagnetic materials. Due to the essential nonlinearity and nonconvex constraint, it is typically solved numerically. In this paper, we developed a finite volume…

Numerical Analysis · Mathematics 2026-05-18 Yunjie Gong , Jingrun Chen , Rui Du , Panchi Li

The aim of this work is to present the details of the finite element approach we developed for solving the Landau-Lifschitz-Gilbert equations in order to be able to treat problems involving complex geometries. There are several…

Other Condensed Matter · Physics 2008-03-31 Helga Szambolics , Liliana-Daniela Buda , Jean-Christophe Toussaint , Olivier Fruchart

We consider the Landau-Lifshitz-Gilbert equation (LLG), which models time-dependent micromagnetic phenomena. We analyze a fully discrete scheme that combines first-order finite elements in space with a BDF2 method in time. The method…

Numerical Analysis · Mathematics 2026-05-07 Michele Aldé , Dirk Praetorius , Michael Feischl

Thermal fluctuations play an increasingly important role in micromagnetic research relevant for various biomedical and other technological applications. Until now, it was deemed necessary to use a time stepping algorithm with a fixed time…

Mesoscale and Nanoscale Physics · Physics 2020-12-24 J. Leliaert , J. Mulkers , J. De Clercq , A. Coene , M. Dvornik , B. Van Waeyenberge

For the Landau--Lifshitz--Gilbert (LLG) equation of micromagnetics we study linearly implicit backward difference formula (BDF) time discretizations up to order $5$ combined with higher-order non-conforming finite element space…

Numerical Analysis · Mathematics 2020-03-23 Georgios Akrivis , Michael Feischl , Balázs Kovács , Christian Lubich

Based on lowest-order finite elements in space, we consider the numerical integration of the Landau-Lifschitz-Gilbert equation (LLG). The dynamics of LLG is driven by the so-called effective field which usually consists of the exchange…

Numerical Analysis · Mathematics 2018-05-04 Dirk Praetorius , Michele Ruggeri , Bernhard Stiftner

High-fidelity numerical simulation serves as a cornerstone for exploring magnetization dynamics in micromagnetics. This work introduces a novel third-order temporally accurate and stable numerical scheme for the Landau-Lifshitz-Gilbert…

Mathematical Physics · Physics 2025-11-27 Changjian Xie

We consider the numerical approximation of the inertial Landau-Lifshitz-Gilbert equation (iLLG), which describes the dynamics of the magnetization in ferromagnetic materials at subpicosecond time scales. We propose and analyze two fully…

Numerical Analysis · Mathematics 2021-10-07 Michele Ruggeri

Energy minimization algorithms, such as graph cuts, enable the computation of the MAP solution under certain probabilistic models such as Markov random fields. However, for many computer vision problems, the MAP solution under the model is…

Computer Vision and Pattern Recognition · Computer Science 2013-07-31 Yongsub Lim , Kyomin Jung , Pushmeet Kohli

We consider the numerical approximation of a continuum model of antiferromagnetic and ferrimagnetic materials. The state of the material is described in terms of two unit-length vector fields, which can be interpreted as the magnetizations…

Numerical Analysis · Mathematics 2023-12-11 Hywel Normington , Michele Ruggeri
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