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, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_99966_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">\(\vert \textbf{m} \vert = m = 1\)</EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_99966_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(m=1\)</EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_99966_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu\)</EquationSource> </InlineEquation>MAG standard problem #4. For this problem the method provides a somewhat faster solution which is of comparable accuracy to other micromagnetic simulation software.

Stabilisation of the Landau-Lifshitz-Gilbert equation for numerical solution via standard methods

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2025年5月6日

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André E. Botha

摘要

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, \(\vert \textbf{m} \vert = m = 1\) . 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 \(m=1\) . 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 \(\mu\) MAG standard problem #4. For this problem the method provides a somewhat faster solution which is of comparable accuracy to other micromagnetic simulation software.