Botha André E
Department of Physics, Science Campus, University of South Africa, Florida Park, Johannesburg, 1710, South Africa.
Sci Rep. 2025 May 6;15(1):15775. doi: 10.1038/s41598-025-99966-x.
Landau and Lifshitz developed their phenomenological equation for the magnetisation dynamics in ferromagnetic solids almost a century ago. At any specific time and position within the solid, the equation describes the rotations of the magnetisation vector under the influence of the effective magnetic field, i.e. the field 'felt' by the magnetisation. The effective field may include, for example, an applied (external) magnetic field as well as internal contributions from the exchange interaction and anisotropy. In 1951 the form of the damping term was modified by Gilbert, and the resulting (mathematically equivalent) equation became known as the Landau-Lifshitz-Gilbert equation. With the rapid increase in computational speed and accessibility during subsequent years, and the growth of computational solid state physics and material science, numerical simulations based on the Landau-Lifshitz-Gilbert equation have greatly assisted with the development of certain technological applications, e.g. hard disk drives. In the present work we review some of the challenges associated with solving this important equation, numerically. In particular, we address the challenge of accurately conserving the magnitude of the magnetisation vector, [Formula: see text]. Usually, if the equation is solved in Cartesian coordinates via an explicit numerical method, m grows or contracts linearly in proportion to the integration time. Here, we show that a new phenomenological term, in the normal form of the supercritical pitchfork bifurcation, can be added to the LLG equation to conserve m numerically, without otherwise affecting the dynamics of the original equation. The additional term stabilises the numerical solution by attracting it to the stable fixed point at [Formula: see text]. Numerical results from seven different solvers are compared to evaluate the effects and efficiency of the additional term. We find that it permits the use of standard, explicit solvers, such as the classic forth-order Runge-Kutta method, to solve the LLG equation more efficiently than pseudo-symplectic or implicit methods, while conserving m to the same accuracy. A Python 3 implementation of the method is provided to solve and compare the μMAG standard problem #4. For this problem the method provides a somewhat faster solution which is of comparable accuracy to other micromagnetic simulation software.
近一个世纪前，朗道（Landau）和栗弗席兹（Lifshitz）推导出了描述铁磁固体中磁化动力学的唯象方程。在固体中的任何特定时间和位置，该方程描述了在有效磁场影响下磁化矢量的旋转，即磁化所“感受”到的场。有效场可能包括例如外加（外部）磁场以及来自交换相互作用和各向异性的内部贡献。1951年，吉尔伯特（Gilbert）修改了阻尼项的形式，所得（数学上等效）方程被称为朗道 - 栗弗席兹 - 吉尔伯特方程。随着随后几年计算速度的快速提升和计算的可及性增加，以及计算固态物理和材料科学的发展，基于朗道 - 栗弗席兹 - 吉尔伯特方程的数值模拟极大地推动了某些技术应用的发展，例如硬盘驱动器。在本工作中，我们回顾了数值求解这个重要方程所面临的一些挑战。特别地，我们解决了精确守恒磁化矢量大小(\vert\vec{m}\vert)的挑战。通常，如果通过显式数值方法在笛卡尔坐标系中求解该方程，(\vec{m})会与积分时间成线性比例地增长或收缩。在此，我们表明可以将一个处于超临界叉形分岔标准形式的新唯象项添加到朗道 - 栗弗席兹 - 吉尔伯特方程中，以在数值上守恒(\vec{m})，而不会以其他方式影响原方程的动力学。该项通过将数值解吸引到(\vert\vec{m}\vert)处的稳定不动点来稳定数值解。比较了七种不同求解器的数值结果，以评估该项的效果和效率。我们发现，它允许使用标准的显式求解器，如经典的四阶龙格 - 库塔方法，比伪辛方法或隐式方法更有效地求解朗道 - 栗弗席兹 - 吉尔伯特方程，同时以相同的精度守恒(\vec{m})。提供了该方法的Python 3实现，用于求解和比较微磁学标准问题#4。对于这个问题，该方法提供了一个速度稍快的解，其精度与其他微磁学模拟软件相当。
