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Enhanced second-order Gauss-Seidel projection methods for the Landau-Lifshitz equation

Panchi Li, Xiao-Ping Wang

TL;DR

This work advances numerical micromagnetics by introducing two second-order Gauss-Seidel projection methods for the Landau-Lifshitz equation. By replacing the Laplacian with a biharmonic-based operator and coupling it with a semi-implicit BDF2 scheme, the authors obtain schemes that are as cheap as solving a scalar biharmonic problem while delivering second-order accuracy. Scheme A is unconditionally stable, whereas Scheme B is conditionally stable under a CFL limit of 0.25; both are complemented by an effective projection step to maintain $|oldsymbol{m}|=1$. Numerical experiments across 1D, 2D, and 3D settings confirm second-order convergence in time and space and demonstrate unconditional stability for micromagnetic simulations, highlighting the practical impact for efficient LL dynamics modeling.

Abstract

The dynamics of magnetization in ferromagnetic materials are modeled by the Landau-Lifshitz equation, which presents significant challenges due to its inherent nonlinearity and non-convex constraint. These complexities necessitate efficient numerical methods for micromagnetics simulations. The Gauss-Seidel Projection Method (GSPM), first introduced in 2001, is among the most efficient techniques currently available. However, existing GSPMs are limited to first-order accuracy. This paper introduces two novel second-order accurate GSPMs based on a combination of the biharmonic equation and the second-order backward differentiation formula, achieving computational complexity comparable to that of solving the scalar biharmonic equation implicitly. The first proposed method achieves unconditional stability through Gauss-Seidel updates, while the second method exhibits conditional stability with a Courant-Friedrichs-Lewy constant of 0.25. Through consistency analysis and numerical experiments, we demonstrate the efficacy and reliability of these methods. Notably, the first method displays unconditional stability in micromagnetics simulations, even when the stray field is updated only once per time step.

Enhanced second-order Gauss-Seidel projection methods for the Landau-Lifshitz equation

TL;DR

This work advances numerical micromagnetics by introducing two second-order Gauss-Seidel projection methods for the Landau-Lifshitz equation. By replacing the Laplacian with a biharmonic-based operator and coupling it with a semi-implicit BDF2 scheme, the authors obtain schemes that are as cheap as solving a scalar biharmonic problem while delivering second-order accuracy. Scheme A is unconditionally stable, whereas Scheme B is conditionally stable under a CFL limit of 0.25; both are complemented by an effective projection step to maintain . Numerical experiments across 1D, 2D, and 3D settings confirm second-order convergence in time and space and demonstrate unconditional stability for micromagnetic simulations, highlighting the practical impact for efficient LL dynamics modeling.

Abstract

The dynamics of magnetization in ferromagnetic materials are modeled by the Landau-Lifshitz equation, which presents significant challenges due to its inherent nonlinearity and non-convex constraint. These complexities necessitate efficient numerical methods for micromagnetics simulations. The Gauss-Seidel Projection Method (GSPM), first introduced in 2001, is among the most efficient techniques currently available. However, existing GSPMs are limited to first-order accuracy. This paper introduces two novel second-order accurate GSPMs based on a combination of the biharmonic equation and the second-order backward differentiation formula, achieving computational complexity comparable to that of solving the scalar biharmonic equation implicitly. The first proposed method achieves unconditional stability through Gauss-Seidel updates, while the second method exhibits conditional stability with a Courant-Friedrichs-Lewy constant of 0.25. Through consistency analysis and numerical experiments, we demonstrate the efficacy and reliability of these methods. Notably, the first method displays unconditional stability in micromagnetics simulations, even when the stray field is updated only once per time step.

Paper Structure

This paper contains 14 sections, 37 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. LL equation
  3. GSPMs for the LL equation
  4. Scheme A
  5. Scheme B
  6. Spatial discretization
  7. Numerical experiments
  8. Numerical accuracy
  9. 1D case
  10. 2D case without source
  11. 3D case
  12. The stability of second-order GSPMs
  13. An unconditionally stable second-order GSPM for micromagnetics simulations
  14. Conclusions

Figures (2)

  • Figure 1: The convergence rate of Scheme A and Scheme B for the full LL equation without source term. The slope of the fitted solid line is 1.91.
  • Figure 2: The simulation of the full LL equation without external fields. Here the magnetization on the centered slice of the material in the $xy$ plane is depicted.

Theorems & Definitions (1)

  • Remark 1