Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a class of higher-order length preserving and energy decreasing IMEX schemes for the Landau-Lifshitz equation

Xiaoli Li, Nan Zheng, Jie Shen

TL;DR

This work addresses stable, high-accuracy time discretization for the Landau-Lifshitz equation with the unit-length constraint. It introduces high-order IMEX-GSAV schemes that couple a generalized SAV with a projection to enforce $|\mathbf{m}|=1$ while ensuring a modified energy is unconditionally decreasing. The authors prove rigorous error estimates up to order five in semi-discrete form, and show that semi-implicit treatment of the nonlinear exchange term can relax time-step constraints. Numerical experiments with spectral spatial discretization corroborate the theoretical rates, demonstrate energy dissipation and length preservation, and illustrate the method's robustness across known and unknown exact solutions, as well as blow-up behavior in the model. These results provide a computationally efficient, theoretically solid framework for simulating micromagnetic dynamics with strong stability and accuracy guarantees.

Abstract

We construct new higher-order implicit-explicit (IMEX) schemes using the generalized scalar auxiliary variable (GSAV) approach for the Landau-Lifshitz equation. These schemes are linear, length preserving and only require solving one elliptic equation with constant coefficients at each time step. We show that numerical solutions of these schemes are uniformly bounded without any restriction on the time step size, and establish rigorous error estimates in $l^{\infty}(0,T;H^1(Ω)) \bigcap l^{2}(0,T;H^2(Ω))$ of orders 1 to 5 in a unified framework.

On a class of higher-order length preserving and energy decreasing IMEX schemes for the Landau-Lifshitz equation

TL;DR

This work addresses stable, high-accuracy time discretization for the Landau-Lifshitz equation with the unit-length constraint. It introduces high-order IMEX-GSAV schemes that couple a generalized SAV with a projection to enforce while ensuring a modified energy is unconditionally decreasing. The authors prove rigorous error estimates up to order five in semi-discrete form, and show that semi-implicit treatment of the nonlinear exchange term can relax time-step constraints. Numerical experiments with spectral spatial discretization corroborate the theoretical rates, demonstrate energy dissipation and length preservation, and illustrate the method's robustness across known and unknown exact solutions, as well as blow-up behavior in the model. These results provide a computationally efficient, theoretically solid framework for simulating micromagnetic dynamics with strong stability and accuracy guarantees.

Abstract

We construct new higher-order implicit-explicit (IMEX) schemes using the generalized scalar auxiliary variable (GSAV) approach for the Landau-Lifshitz equation. These schemes are linear, length preserving and only require solving one elliptic equation with constant coefficients at each time step. We show that numerical solutions of these schemes are uniformly bounded without any restriction on the time step size, and establish rigorous error estimates in of orders 1 to 5 in a unified framework.
Paper Structure (10 sections, 8 theorems, 95 equations, 6 figures)

This paper contains 10 sections, 8 theorems, 95 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Reformulations for the Landau-Lifshitz equation and preliminaries
  3. Norm preserving and energy decreasing IMEX-GSAV schemes
  4. Error estimate
  5. Semi-implicit discretization for the term $\beta \textbf{m}\times \Delta \textbf{m}$
  6. Numerical experiments
  7. Convergence rate with a known exact solution
  8. Convergence rate with an unknown exact solution
  9. Phenomenon of blow up
  10. Concluding remarks

Key Result

Lemma 2.1

(Hölder inequality) \newlabellem: Holder inequality Let $p, q, s >0$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{s}=1$. Then for vector functions $\textbf{u} \in \textbf{L}^p(\Omega)$, $\textbf{v}\in \textbf{L}^q(\Omega)$, and scalar function $w \in L^s(\Omega)$, we have

Figures (6)

  • Figure 6.1: Numerical convergence rate of the first- to fourth-order schemes with $\beta=0$ in Example 1.
  • Figure 6.2: Numerical convergence rate of the first- to fourth-order schemes with $\beta=0.5$ in Example 1.
  • Figure 6.3: Numerical convergence rate of the first- to fourth-order schemes in Example 2.
  • Figure 6.4: Numerical magnetization m (projected on $x_1x_2$-plane) with $N=128$.
  • Figure 6.5: Numerical magnetization m around the origin with $N=128$.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • Theorem 4.2
  • proof
  • ...and 4 more