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Highly efficient norm preserving numerical schemes for micromagnetic energy minimization based on SAV method

Jiayun He, Lei Yang, Jiajun Zhan

TL;DR

Ground-state computation in micromagnetics is formulated as a constrained Gibbs energy minimization with $|m|=1$. The authors develop two scalar auxiliary variable (SAV) schemes, SAV1 and SAV2, that transform nonlinear terms to linear-implicit updates and enforce the norm constraint via an implicit projection. Magnetostatic energy is incorporated through a scalar variable $r$, enabling two constant-coefficient linear solves per step, accelerated by the Discrete Cosine Transform. Numerical experiments show that the SAV schemes deliver energy-dissipative, stable, and efficient performance with close agreement to reference LLG-based solutions, enabling faster ground-state computations in micromagnetic simulations.

Abstract

In this paper, two efficient and magnetization norm preserving numerical schemes based on the scalar auxiliary variable (SAV) method are developed for calculating the ground state in micromagnetic structures. The first SAV scheme is based on the original SAV method for the gradient flow model, while the second scheme features an updated scalar auxiliary variable to better align with the associated energy. To address the challenging constraint of pointwise constant magnetization length, an implicit projection method is designed, and verified by both SAV schemes. Both proposed SAV schemes partially preserve energy dissipation and exhibit exceptional efficiency, requiring two linear systems with constant coefficients to be solved. The computational efficiency is further enhanced by applying the Discrete Cosine Transform during the solving process. Numerical experiments demonstrate that our SAV schemes outperform commonly used numerical methods in terms of both efficiency and stability.

Highly efficient norm preserving numerical schemes for micromagnetic energy minimization based on SAV method

TL;DR

Ground-state computation in micromagnetics is formulated as a constrained Gibbs energy minimization with . The authors develop two scalar auxiliary variable (SAV) schemes, SAV1 and SAV2, that transform nonlinear terms to linear-implicit updates and enforce the norm constraint via an implicit projection. Magnetostatic energy is incorporated through a scalar variable , enabling two constant-coefficient linear solves per step, accelerated by the Discrete Cosine Transform. Numerical experiments show that the SAV schemes deliver energy-dissipative, stable, and efficient performance with close agreement to reference LLG-based solutions, enabling faster ground-state computations in micromagnetic simulations.

Abstract

In this paper, two efficient and magnetization norm preserving numerical schemes based on the scalar auxiliary variable (SAV) method are developed for calculating the ground state in micromagnetic structures. The first SAV scheme is based on the original SAV method for the gradient flow model, while the second scheme features an updated scalar auxiliary variable to better align with the associated energy. To address the challenging constraint of pointwise constant magnetization length, an implicit projection method is designed, and verified by both SAV schemes. Both proposed SAV schemes partially preserve energy dissipation and exhibit exceptional efficiency, requiring two linear systems with constant coefficients to be solved. The computational efficiency is further enhanced by applying the Discrete Cosine Transform during the solving process. Numerical experiments demonstrate that our SAV schemes outperform commonly used numerical methods in terms of both efficiency and stability.

Paper Structure

This paper contains 11 sections, 1 theorem, 39 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Micromagnetic flow model
  3. Numerical schemes
  4. Numerical schemes for flow equation
  5. SAV discrete schemes
  6. Energy dissipation of discrete SAV schemes
  7. Analysis of computational cost
  8. Numerical experiments
  9. Effectiveness and convergence
  10. Efficiency
  11. Conclusion

Key Result

Theorem 3.1

Assume $\boldsymbol{m}^{*}$ is obtained by SAV1 scheme Eqn:SAVTD or SAV2 scheme Eqn:RTUT. If the following two conditions are satisfied, then the energy dissipation inequality holds.

Figures (10)

  • Figure 1: (a) Initial magnetization \ref{['Eqn:IMDiamond']}; (b) the corresponding steady-state named as diamond state. The color mapping indicates the angle between the magnetization vector and the $x$-axis.
  • Figure 2: Comparison of energies as functions of time by applying SAV1 scheme with initial magnetization \ref{['Eqn:IMDiamond']} and $\Delta t = 10^{-12}$.
  • Figure 3: Evolution of the original and modified energies over time under the SAV1 scheme with time steps of (a) $\Delta t = 10^{-13}$ and (b) $\Delta t = 10^{-14}$.
  • Figure 4: Comparison of time-dependent total energy profiles for the SAV2 scheme with initial state \ref{['Eqn:IMDiamond']} across a range of time steps ($\Delta t = 10^{-12}$, $10^{-13}$, and $10^{-14}$).
  • Figure 5: (a) Initial magnetization \ref{['Eqn:IMSingleCrossTie']}; (b) the corresponding steady-state named as single cross-tie state. The color mapping indicates the angle between the magnetization vector and the $x$-axis.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Example 4.1