An Efficient Energy Stable Structure Preserving Method for The Landau-Lifshitz Equation

TL;DR

The paper tackles the unit-length constraint and time-step stability in simulating the Landau-Lifshitz-Gilbert equation for micromagnetics. It introduces a projection-free, structure-preserving time integrator that combines Gauss-Seidel iterations, a double-diffusion refinement, and Crank-Nicolson-type stepping to preserve $||\mathbf{m}||=1$ while maintaining energy stability, without explicit projection. The method demonstrates first-order temporal accuracy and second-order spatial accuracy in 1D and 3D tests, along with robust norm preservation and stability under diverse initial conditions. This approach offers a scalable alternative to projection-based schemes for large-scale micromagnetic simulations and sets the stage for further stability analysis and extensions to full LLG models with damping and stray fields.

Abstract

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 explicit integrators leads to a physical time step of sub-pico seconds, which is often two orders of magnitude smaller than the fastest physical time scales. Direct implicit integrators require solving complicated, coupled systems. Another major difficulty with the projection method in this field is the lack of rigorous theoretical guarantees regarding its stability of the projection step. In this paper, we introduce a first order method. Such a method is structure preserving based on a combination of a Gauss-Seidel iteration, a double diffusion iteration and a Crank-Nicolson iteration to preserve the norm constraints.

Figures (12)

• Figure 1: The solution profile using proposed method in 1D given the initial condition $m_0$ without source term, $\alpha=0.01$ and $T=0.1$, $N_x=2000$, $N_t=5$. Left panel: the initial condition is given by $\hbox{\boldmath $m$}_0=\left(\cos(\cos(\pi x))\sin (0), \sin(\cos(\pi x))\sin (0), \cos (0)\right)^T$; Right panel: the initial condition is given by $\hbox{\boldmath $m$}_0=\left(\cos(\cos(\pi x))\sin (0.01), \sin(\cos(\pi x))\sin (0.01), \cos (0.01)\right)^T$.
• Figure 2: The solution profile using GSPM method in 1D given the initial condition $m_0$ without source term, $\alpha=0.01$ and $T=0.1$, $N_x=2000$, $N_t=5$. Left panel: the initial condition is given by $\hbox{\boldmath $m$}_0=\left(\cos(\cos(\pi x))\sin (0), \sin(\cos(\pi x))\sin (0), \cos (0)\right)^T$; Right panel: the initial condition is given by $\hbox{\boldmath $m$}_0=\left(\cos(\cos(\pi x))\sin (0.01), \sin(\cos(\pi x))\sin (0.01), \cos (0.01)\right)^T$.
• 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 GSPM; Bottom row with proposed method. Initial condition given: $\hbox{\boldmath $m$}_0=[\cos(\cos(\pi x))\sin(x+t),\sin(\cos(\pi x))\sin(x+t),\cos(x+t)]$ with $t=T0=0$.
• Figure 5: 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 GSPM; 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 $t=T0=0$.