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We present efficient numerical methods for the simulation of small magnetization oscillations in three-dimensional micromagnetic systems. Magnetization dynamics is described by the Landau-Lifshitz-Gilbert (LLG) equation, linearized in the…

Computational Physics · Physics 2023-01-18 Massimiliano d'Aquino , Riccardo Hertel

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to…

Statistical Mechanics · Physics 2014-08-27 Federico Romá , Leticia F. Cugliandolo , Gustavo S. Lozano

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

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

We present a general finite element linearized Landau-Lifshitz-Gilbert equation (LLGE) solver for magnetic systems under weak time-harmonic excitation field. The linearized LLGE is obtained by assuming a small deviation around the…

Applied Physics · Physics 2022-10-27 Zhuonan Lin , Vitaliy Lomakin

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

We consider the numerical approximation of a nonlinear system of partial differential equations modeling magnetostriction in the small-strain regime consisting of the Landau--Lifshitz--Gilbert equation for the magnetization and the…

Numerical Analysis · Mathematics 2026-04-01 Martin Kružík , Hywel Normington , Michele Ruggeri

The numerical approximation for the Landau-Lifshitz equation, the dynamics of magnetization in a ferromagnetic material, is taken into consideration. This highly nonlinear equation, with a non-convex constraint, has several equivalent…

Analysis of PDEs · Mathematics 2019-07-05 Jingrun Chen , Cheng Wang , Changjian Xie

Magnetization dynamics in ferromagnetic materials is modeled by the Landau-Lifshitz (LL) equation, a nonlinear system of partial differential equations. Among the numerical approaches, semi-implicit schemes are widely used in the…

Numerical Analysis · Mathematics 2023-12-27 Yan Gui , Cheng Wang , Jingrun Chen

Simulations of magnetization dynamics in a multiscale environment enable rapid evaluation of the Landau-Lifshitz-Gilbert equation in a mesoscopic sample with nanoscopic accuracy in areas where such accuracy is required. We have developed a…

Computational Physics · Physics 2016-11-28 Andrea De Lucia , Benjamin Krüger , Oleg A. Tretiakov , Mathias Kläui

The stochastic Landau--Lifshitz--Gilbert (LLG) equation describes the behaviour of the magnetization under the influence of the effective field consisting of random fluctuations. We first reformulate the equation into an equation the…

Numerical Analysis · Mathematics 2015-02-05 Beniamin Goldys , Kim-Ngan Le , Thanh Tran

The Landau-Lifshitz-Gilbert equation yields a mathematical model to describe the evolution of the magnetization of a magnetic material, particularly in response to an external applied magnetic field. It allows one to take into account…

Numerical Analysis · Mathematics 2021-06-15 Tram Thi Ngoc Nguyen , Anne Wald

The classical Landau-Lifshitz-Gilbert (LLG) equation has long served as a cornerstone for modeling magnetization dynamics in magnetic systems, yet its classical nature limits its applicability to inherently quantum phenomena such as…

Quantum Physics · Physics 2025-06-25 Vahid Azimi-Mousolou , Davoud Mirzaei

The Landau-Lifshitz-Gilbert (LLG) equation is widely used to describe magnetization dynamics. We develop a unified framework of the microscopic LLG equation based on the nonequilibrium Green's function formalism. We present a unified…

Mesoscale and Nanoscale Physics · Physics 2023-10-16 Fuming Xu , Gaoyang Li , Jian Chen , Zhizhou Yu , Lei Zhang , Baigeng Wang , Jian Wang

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…

Numerical Analysis · Mathematics 2016-12-21 Eugenia Kim , Konstantin Lipnikov

We consider the numerical solution of the inertial version of Landau-Lifshitz-Gilbert equation (iLLG), which describes high-frequency nutation on top of magnetization precession due to angular momentum relaxation. The iLLG equation defines…

Computational Physics · Physics 2023-10-16 Massimiliano d'Aquino , Salvatore Perna , Claudio Serpico

This work proposes and analyzes a fully discrete numerical scheme for solving the Landau-Lifshitz-Gilbert (LLG) equation, which achieves fourth-order spatial accuracy and third-order temporal accuracy.Spatially, fourth-order accuracy is…

Numerical Analysis · Mathematics 2025-10-30 Changjian Xie , Cheng Wang

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

We propose a three-dimensional micromagnetic model that dynamically solves the Landau-Lifshitz-Gilbert equation coupled to the full spin-diffusion equation. In contrast to previous methods, we solve for the magnetization dynamics and the…

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
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