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The dynamics of magnetization near a stable equilibrium in ferromagnetic nanomagnets are examined within the Landau--Lifshitz--Gilbert (LLG) framework. For a small angle precession, the dependence of ferromagnetic resonance (FMR) frequency,…

The Landau-Lifshitz equation is a coupled set of nonlinear partial differential equations that describes the dynamics of magnetization in a ferromagnet. This equation has an infinite number of stable equilibria. Steering the system from one…

Analysis of PDEs · Mathematics 2016-05-31 Amenda Chow , Kirsten A. Morris

Magnetization dynamics is commonly described by the stochastic Landau-Lifshitz-Gilbert (LLG) equation. On picosecond timescales, inertial and open-system extensions of the LLG equation are necessary to interpret recent experiments. We show…

Statistical Mechanics · Physics 2026-02-20 Felix Hartmann , Finja Tietjen , R. Matthias Geilhufe , Janet Anders

Spin currents act on ferromagnets by exerting a torque on the magnetisation. This torque is modelled by appending additional terms to the Landau-Lifshitz-Gilbert equation motivating the study of the non-homogeneous Landau-Lifshitz-Gilbert…

Analysis of PDEs · Mathematics 2024-07-16 Noah Vinod , Thanh Tran

The Landau-Lifshitz-Gilbert (LLG) equation, regarded as a gradient flow with manifold constraint, is the fundamental model describing magnetization dynamics in ferromagnetic materials. It is well known that the normalized tangent plane…

Numerical Analysis · Mathematics 2026-02-10 Binghong Li , Xiaoli Li , Cheng Wang , Jiang Yang

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

This paper investigates the time-optimal control problem for the Landau-Lifshitz-Bloch (LLB) equation, a macroscopic model that characterizes magnetization dynamics in ferromagnetic materials across a wide temperature range, including near…

Optimization and Control · Mathematics 2025-12-22 Sidhartha Patnaik , Kumarasamy Sakthivel

The tangent plane scheme is a time-marching scheme for the numerical solution of the nonlinear parabolic Landau-Lifshitz-Gilbert equation (LLG), which describes the time evolution of ferromagnetic configurations. Exploiting the geometric…

Numerical Analysis · Mathematics 2020-03-03 Johannes Kraus , Carl-Martin Pfeiler , Dirk Praetorius , Michele Ruggeri , Bernhard Stiftner

A non-conventional finite element formalism is proposed to solve the dynamic Landau-Lifshitz-Gilbert micromagnetic equations. Two bidimensional test problems are treated to estimate the validity and the accuracy of this finite element…

The Landau--Lifshitz--Baryakhtar (LLBar) and the Landau--Lifshitz--Bloch (LLBloch) equations are nonlinear vector-valued PDEs which arise in the theory of micromagnetics to describe the dynamics of magnetic spin field in a ferromagnet at…

Numerical Analysis · Mathematics 2025-05-27 Agus L. Soenjaya

Here we consider the magnetization dynamics in ferromagnetic nanoparticles or films driven by external magnetic field pulses directed transverse to the initial equilibrium state. The excitation pulse drives large-angle ultra-fast…

Dynamical Systems · Mathematics 2024-10-23 Alessandro Fortunati , Massimiliano d'Aquino , Claudio Serpico

We propose a $\theta$-linear scheme for the numerical solution of the quasi-static Maxwell-Landau-Lifshitz-Gilbert (MLLG) equations. Despite the strong nonlinearity of the Landau-Lifshitz-Gilbert equation, the proposed method results in a…

Numerical Analysis · Mathematics 2012-12-17 Kim-Ngan Le , T. Tran

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 ultra-fast magnetisation relaxation rates during the laser-induced magnetisation process are analyzed in terms of the Landau-Lifshitz-Bloch (LLB) equation for different values of spin $S$. The LLB equation is equivalent in the limit $S…

Materials Science · Physics 2015-05-20 U. Atxitia , O. Chubykalo-Fesenko

We study the effect of an elliptically polarized magnetic field on a system of non-interacting, single-domain ferromagnetic nanoparticles characterized by a uniform distribution of easy axis directions. Our main goal is to determine the…

Mesoscale and Nanoscale Physics · Physics 2016-07-06 S. I. Denisov , T. V. Lyutyy , B. O. Pedchenko , O. M. Hryshko

The Landau--Lifshitz--Bloch (LLB) equation is a well-established micromagnetic model for describing magnetisation dynamics in ferromagnets at elevated temperatures. In this paper, we propose and analyse two fully discrete, conforming finite…

Numerical Analysis · Mathematics 2026-03-04 Agus L. Soenjaya

We consider the one-dimensional Landau-Lifshitz-Gilbert (LLG) equation, a model describing the dynamics for the spin in ferromagnetic materials. Our main aim is the analytical study of the bi-parametric family of self-similar solutions of…

Analysis of PDEs · Mathematics 2017-01-13 Susana Gutiérrez , André de Laire

The Landau--Lifshitz equation describes the dynamics of magnetization inside a ferromagnet. This equation is nonlinear and has an infinite number of stable equilibria. It is desirable to control the system from one equilibrium to another. A…

Optimization and Control · Mathematics 2015-09-29 Amenda Chow , Kirsten A. Morris

The Landau-Lifshitz (L-L) equation describing the time dependence of the magnetisation vector is numerically integrated fully without any simplifying assumptions in the time domain and the magnetisation time series obtained is Fourier…

Materials Science · Physics 2009-09-25 C. Tannous , A. Fessant , J. Gieraltowski , J. P. Jay , D. Spenato , J. Langlois , H. Le Gall

Numerical integration of the Landau-Lifshitz-Gilbert equation with thermal fluctuations is used to study the dynamic response of single-domain nanomagnets to rapid changes in the applied magnetic field. The simulation can resolve…

Materials Science · Physics 2007-05-23 G. Brown , M. A. Novotny , P. A. Rikvold