Article Contents
Article Contents

# A Gauss-Seidel projection method with the minimal number of updates for the stray field in micromagnetics simulations

• *Corresponding authors: Rui Du and Jingrun Chen

P. Li was supported by the Postgraduate Research & Practice Innovation Program of Jiangsu Province via grant KYCX20_2711, R. Du was supported by NSFC via grant 11501399, and J. Chen was supported by NSFC via grant 11971021

• Magnetization dynamics in magnetic materials is often modeled by the Landau-Lifshitz equation, which is solved numerically in general. In micromagnetics simulations, the computational cost relies heavily on the time-marching scheme and the evaluation of the stray field. In this work, we propose a new method, dubbed as GSPM-BDF2, by combining the advantages of the Gauss-Seidel projection method (GSPM) and the second-order backward differentiation formula scheme (BDF2). Like GSPM, this method is first-order accurate in time and second-order accurate in space, and it is unconditionally stable with respect to the damping parameter. Remarkably, GSPM-BDF2 updates the stray field only once per time step, leading to an efficiency improvement of about $60\%$ compared with the state-of-the-art of GSPM for micromagnetics simulations. For Standard Problems #4 and #5 from National Institute of Standards and Technology, GSPM-BDF2 reduces the computational time over the popular software OOMMF by $82\%$ and $96\%$, respectively. Thus, the proposed method provides a more efficient choice for micromagnetics simulations.

Mathematics Subject Classification: Primary: 35Q99, 65Z05, 65M06.

 Citation:

• Figure 1.  Simulation results of the magnetization on the centered slice of the material in the $xy$ plane. Left column: a color plot of the angle between the in-plane magnetization and the $x$-axis; Right column: an arrow plot of the in-plane magnetization; Top row: GSPM with one update of the stray field; Second row: GSPM with three updates of the stray field; Bottom row: GSPM-BDF2

Figure 2.  Magnetization profile of the initial s-state. This is generated for a random initial state under a magnetic field $100\;\mathrm{mT}$ along the $[1, 1, 1]$ direction with a successive reduction to $0$

Figure 3.  Left column: dynamics of the spatially averaged magnetization on the coarse mesh under the external fields; Right column: comparison of the averaged $\langle m_y\rangle$ on the two different meshes. Top row: Field Ⅰ; Bottom row: Field Ⅱ

Figure 4.  Magnetization profile when the averaged $\langle m_x\rangle = 0$ under the external fields. The color map is given by the $z$-component. Top row: Field Ⅰ; Bottom row: Field Ⅱ

Figure 5.  Magnetization dynamics and the final state for four sets of parameters. Left column: dynamics of the spatially averaged magnetization with the result of D. G. Porter as the reference; Right column: final state colored by the $x$-component. Top row: Case 1; Second row: Case 2; Third row: Case 3; Bottom row: Case 4

Figure 6.  Magnetization dynamics and the final state in the case of $\xi = 0.5$ and $bJ = 72.45 \;\mathrm{m}/\mathrm{s}$. Left: dynamics of the spatially averaged magnetization with the result of G. Finocchio et al. as the reference; Right: final state colored by the $x$-component. The $x$-component of the spatially averaged magnetization $\langle M_x\rangle$ at $10\;\mathrm{ns}$ is $-1.43\times10^5\;\mathrm{A}/\mathrm{m}$

Table 1.  Main computational costs of GSPM and GSPM-BDF2 per time step

 #(linear systems of equations) (dofs) #(stray field updates) (dofs) GSPM 5 ($N$) 3 ($3N$) GSPM-BDF2 5 ($N$) 1 ($3N$)

Table 2.  Convergence rates in terms of $\Delta t$ and $\Delta x$ for Example 1 (1D)

 Temporal accuracy $\Delta t$ T/1000 T/2000 T/4000 T/8000 order GSPM 3.42e-07 1.71e-07 0.86e-07 0.43e-09 0.99 GSPM-BDF2 3.42e-07 1.71e-07 0.86e-07 0.43e-09 0.99 Spatial accuracy $\Delta x$ 1/20 1/40 1/80 1/160 order GSPM 1.29e-04 0.39e-04 0.11e-04 0.03e-04 1.82 GSPM-BDF2 1.29e-04 0.39e-04 0.11e-04 0.03e-04 1.82

Table 3.  Convergence rates in terms of $\Delta t$ and $h$ for Example 2 (3D)

 Temporal accuracy $\Delta t$ T/100 T/200 T/400 T/800 order GSPM 1.00e-07 5.00e-08 2.50e-08 1.25e-08 1.00 GSPM-BDF2 1.00e-07 5.00e-08 2.50e-08 1.25e-08 1.00 Spatial accuracy $h$ 1/6 1/8 1/10 1/12 order GSPM 2.91e-14 1.72e-14 1.13e-14 7.92e-15 1.88 GSPM-BDF2 2.91e-14 1.72e-14 1.13e-14 7.92e-15 1.88

Table 4.  Computational costs (in seconds) of GSPM-BDF2 and OOMMF for Standard Problem #4 when the coarse mesh is used

 Standard Problem #4 GSPM-BDF2 OOMMF Saving Field Ⅰ 20.47 115.32 82% Field Ⅱ 20.33 116.41 83%

Table 5.  Computational costs (in seconds) of GSPM-BDF2 and OOMMF for Standard Problem #5

 Parameters GSPM-BDF2 OOMMF Saving Case 1 97.58 2216.85 96% Case 2 97.55 2226.45 96% Case 3 93.91 2229.22 96% Case 4 95.59 2246.40 96%
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