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Error Analysis of Third-Order in Time and Fourth-Order Linear Finite Difference Scheme for Landau-Lifshitz-Gilbert Equation under Large Damping Parameters

Changjian Xie, Cheng Wang

TL;DR

This work develops a fully discrete finite-difference scheme for the Landau-Lifshitz-Gilbert equation that achieves 4th-order spatial accuracy and 3rd-order temporal accuracy. Spatial accuracy is attained via long-stencil finite differences, boundary extrapolation, and a Neumann boundary treatment, while time stepping uses a 3rd-order BDF with implicit diffusion and explicit nonlinear terms, followed by a projection to enforce |m|=1. The authors prove an optimal convergence rate of O(k^3 + h^4) in discrete L2 and H1 norms (under regularity and a large-damping condition α > 7, and k ≤ C h), and validate the theory with 1D and 3D numerical experiments demonstrating the expected spatial and temporal orders. The approach enables high-accuracy, stable simulations for ferromagnetic materials in the large-damping regime, with computational efficiency aided by a constant-coefficient Poisson solve at each step.

Abstract

This work proposes and analyzes a fully discrete numerical scheme for solving the Landau-Lifshitz-Gilbert (LLG) equation, which achieves fourth-order spatial accuracy and third-order temporal accuracy.Spatially, fourth-order accuracy is attained through the adoption of a long-stencil finite difference method, while boundary extrapolation is executed by leveraging a higher-order Taylor expansion to ensure consistency at domain boundaries. Temporally, the scheme is constructed based on the third-order backward differentiation formula (BDF3), with implicit discretization applied to the linear diffusion term for numerical stability and explicit extrapolation employed for nonlinear terms to balance computational efficiency. Notably, this numerical method inherently preserves the normalization constraint of the LLG equation, a key physical property of the system.Theoretical analysis confirms that the proposed scheme exhibits optimal convergence rates under the \(\ell^{\infty}([0,T],\ell^2)\) and \(\ell^2([0,T],H_h^1)\) norms. Finally, numerical experiments are conducted to validate the correctness of the theoretical convergence results, demonstrating good agreement between numerical observations and analytical conclusions.

Error Analysis of Third-Order in Time and Fourth-Order Linear Finite Difference Scheme for Landau-Lifshitz-Gilbert Equation under Large Damping Parameters

TL;DR

This work develops a fully discrete finite-difference scheme for the Landau-Lifshitz-Gilbert equation that achieves 4th-order spatial accuracy and 3rd-order temporal accuracy. Spatial accuracy is attained via long-stencil finite differences, boundary extrapolation, and a Neumann boundary treatment, while time stepping uses a 3rd-order BDF with implicit diffusion and explicit nonlinear terms, followed by a projection to enforce |m|=1. The authors prove an optimal convergence rate of O(k^3 + h^4) in discrete L2 and H1 norms (under regularity and a large-damping condition α > 7, and k ≤ C h), and validate the theory with 1D and 3D numerical experiments demonstrating the expected spatial and temporal orders. The approach enables high-accuracy, stable simulations for ferromagnetic materials in the large-damping regime, with computational efficiency aided by a constant-coefficient Poisson solve at each step.

Abstract

This work proposes and analyzes a fully discrete numerical scheme for solving the Landau-Lifshitz-Gilbert (LLG) equation, which achieves fourth-order spatial accuracy and third-order temporal accuracy.Spatially, fourth-order accuracy is attained through the adoption of a long-stencil finite difference method, while boundary extrapolation is executed by leveraging a higher-order Taylor expansion to ensure consistency at domain boundaries. Temporally, the scheme is constructed based on the third-order backward differentiation formula (BDF3), with implicit discretization applied to the linear diffusion term for numerical stability and explicit extrapolation employed for nonlinear terms to balance computational efficiency. Notably, this numerical method inherently preserves the normalization constraint of the LLG equation, a key physical property of the system.Theoretical analysis confirms that the proposed scheme exhibits optimal convergence rates under the \(\ell^{\infty}([0,T],\ell^2)\) and \(\ell^2([0,T],H_h^1)\) norms. Finally, numerical experiments are conducted to validate the correctness of the theoretical convergence results, demonstrating good agreement between numerical observations and analytical conclusions.

Paper Structure

This paper contains 8 sections, 8 theorems, 68 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Main theoretical results
  3. Fully discretization
  4. Main theorem
  5. A few preliminary estimates
  6. The optimal rate convergence analysis: Proof of Theorem \ref{['cccthm2']}
  7. Numerical examples
  8. Conclusions

Key Result

Theorem 2.1

Let $\hbox{\boldmath $m$}_e \in C^4 ([0,T]; C^0) \cap C^3([0,T]; C^1) \cap L^{\infty}([0,T]; C^6)$ be the exact solution of c1 with the initial data $\hbox{\boldmath $m$}_e ({\hbox{\boldmath $x$}},0)=\hbox{\boldmath $m$}_e ^0({\hbox{\boldmath $x$}})$ and ${\hbox{\boldmath $m$}}_h$ be the numerical s in which the constant $\mathcal{C}>0$ is independent of $k$ and $h$.

Theorems & Definitions (18)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.1: Inner product and $\| \cdot \|_2$ norm
  • Definition 2.2: Discrete $\| \cdot \|_\infty$ norm
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Summation by parts
  • ...and 8 more