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 paper introduces a high-order finite difference scheme for the Landau-Lifshitz-Gilbert equation that achieves fourth-order spatial and third-order temporal accuracy, ensuring stability, physical constraint preservation, and validated convergence through numerical experiments.

Contribution

The work develops a fully discrete, high-order numerical scheme combining fourth-order spatial accuracy with third-order temporal accuracy for the LLG equation, including boundary treatment and stability analysis.

Findings

The scheme achieves optimal convergence rates in key norms.

Numerical experiments confirm theoretical accuracy and stability.

The method preserves the physical normalization constraint of the LLG equation.

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.