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The dynamics of a magnetic moment or spin are of high interest to applications in technology. Dissipation in these systems is therefore of importance for improvement of efficiency of devices, such as the ones proposed in spintronics. A…

Mesoscale and Nanoscale Physics · Physics 2023-08-29 Robin C. Verstraten , Tim Ludwig , Rembert A. Duine , Cristiane Morais Smith

The convergence analysis of a third-order scheme for the highly nonlinear Landau-Lifshitz-Gilbert equation with a non-convex constraint is considered. In this paper, we first present a fully discrete semi-implicit method for solving the…

Numerical Analysis · Mathematics 2025-11-14 Changjian Xie , 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

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

One of the main difficulties in micromagnetics simulation is the norm preserving constraints $\|\mathbf{m}\|=1$ at the continuous or the discrete level. Another difficulty is the stability with the time step constraint. Using standard…

Numerical Analysis · Mathematics 2026-02-12 Changjian Xie , Yingxi Miao , Haocheng Yang

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,…

Magnetization dynamics in magnetic materials is often modeled by the Landau-Lifshitz equation, which is solved numerically in general. In micromagnetic simulations, the computational cost relies heavily on the time-marching scheme and the…

Numerical Analysis · Mathematics 2022-09-09 Panchi Li , Zetao Ma , Rui Du , Jingrun Chen

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 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

In this work, we consider the construction of efficient surrogates for the stochastic version of the Landau-Lifshitz-Gilbert (LLG) equation using model order reduction techniques, in particular, the Reduced Basis (RB) method. The Stochastic…

Numerical Analysis · Mathematics 2025-10-01 Andrea Scaglioni , Michael Feischl , Fernando Henríquez

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

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

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

This paper proposes a decoupled numerical scheme of the time-dependent Ginzburg--Landau equations under the temporal gauge. For the magnetic potential and the order parameter, the discrete scheme adopts the second type Ned${\rm…

Numerical Analysis · Mathematics 2023-07-26 Limin Ma , Zhonghua Qiao

In this paper, the coupled fractional Ginzburg-Landau equations are first time investigated numerically. A linearized implicit finite difference scheme is proposed. The scheme involves three time levels, is unconditionally stable and…

Numerical Analysis · Mathematics 2018-06-01 Dongdong He , Kejia Pan

The current-induced magnetisation dynamics in a ferromagnet at elevated temperatures can be described by the Landau--Lifshitz--Bloch (LLB) equation with spin-torque terms. In this paper, we focus on the regime above the Curie temperature.…

Numerical Analysis · Mathematics 2025-06-30 Agus L. Soenjaya

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…

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

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