Semi-implicit Structure Preserving Method for The Landau-Lifshitz Equation

TL;DR

This work introduces a semi-implicit, structure-preserving discretization for the Landau-Lifshitz-Gilbert equation that avoids explicit projection while maintaining the magnetization constraint $|\boldsymbol{m}|=1$ and providing stable, first-order accuracy in time. Building on a BDF1 framework with one-sided extrapolation and a Crank–Nicolson-type norm-preserving step, the scheme achieves unique solvability and preserves the intrinsic physical structure of the LLG dynamics. Numerical experiments in 1D and 3D demonstrate first-order temporal convergence, second-order spatial accuracy, and robust norm preservation across diverse initial conditions, including comparisons with projection-based methods. The method offers a practical, theoretically favorable alternative for micromagnetic simulations and lays groundwork for rigorous stability analysis and higher-order extensions.

Abstract

A critical challenge inherent to the projection method applied to the Landau-Lifshitz equation is the deficiency of rigorous theoretical justifications for the stability of its projection step. To mitigate this limitation, we introduce a semi-implicit numerical scheme, which is formulated on the basis of the first-order Backward Differentiation Formula (BDF) incorporated with one-sided extrapolation and a Crank-Nicolson-type norm-preserving procedure. This proposed scheme exhibits three fundamental characteristics: structure preservation, numerical stability, and first-order accuracy in time. In practical implementations, the scheme not only ensures stable computation and adheres to the norm constraint but also guarantees the uniqueness of the numerical solution, thereby providing substantial facilitation for the theoretical analysis of the normalizing step.

Figures (6)

• Figure 1: The solution profile using proposed method in 1D given the initial condition $m_0$ with $T0$ specified without source term, $\alpha=0.01$ and $T=0.1$, $N_x=2000$, $N_t=5$.
• Figure 2: The solution profile using BDF1 projection method in 1D given the initial condition $m_0$ with $T0$ specified without source term, $\alpha=0.01$ and $T=0.1$, $N_x=2000$, $N_t=5$.
• Figure 3: The solution profile using GSPM and proposed methods in 3D given the initial condition $m_0$ with initial condition specified without source term, $\alpha=0$ and $T=0.1$, $N_x=N_y=N_z=20$, $N_t=400$. Top row with initial condition; Middle row with GSPM; Bottom row with proposed method. Initial condition given: $\hbox{\boldmath $m$}_0=[\cos(\cos(\pi x))\sin(0.01),\sin(\cos(\pi x))\sin(0.01),\cos(0.01)]$
• Figure 4: The solution profile using GSPM and proposed methods in 3D given the initial condition $m_0$ with initial condition specified without source term, $\alpha=0$ and $T=0.1$, $N_x=N_y=N_z=20$, $N_t=40$. Top row with initial condition; Middle row with BDF1; Bottom row with proposed method. Initial condition given: $\hbox{\boldmath $m$}_0=[\cos(\cos(\cos(\pi x)))\sin(\pi x+t),\sin(\cos(\cos(\pi x)))\sin(\pi x+t),\cos(\pi x+t)]$ with $t=T0=0$.
• Figure 5: Given initial conditions in 3D as $\hbox{\boldmath $m$}_0=[\cos(XYZ)\sin(0.01),\sin(XYZ)\sin(0.01),\cos(0.01)]$ specified without source term, $\alpha=0.01$ and $T=0.1$, $N_x=N_y=N_z=20$, $N_t=40$. Top row: initial condition; Middle row: BDF1 projection method; Bottom row: proposed method.