Second-order semi-implicit projection methods for micromagnetics simulations

Xie, Changjian; Garcı́a-Cervera, Carlos J.; Wang, Cheng; Zhou, Zhennan; Chen, Jingrun

doi:10.1016/j.jcp.2019.109104

Cited by 26 publications

(22 citation statements)

References 39 publications

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“…A semi-implicit numerical scheme has been proposed in [29], and used in the numerical simulation for small damping parameter models. In more details, semi-implicit approximations are applied to the two nonlinear terms, namely m × ∆m and m × (m × ∆m), in which ∆m is treated implicitly, while the coefficient variables are explicitly updated via an extrapolation formula.…”

Section: 2mentioning

confidence: 99%

Convergence Analysis of A Second-order Accurate, Linear Numerical Scheme for The Landau-Lifshitz Equation with Large Damping Parameters

Cai¹,

Chen²,

Wang³

et al. 2021

Preprint

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A second order accurate, linear numerical method is analyzed for the Landau-Lifshitz equation with large damping parameters. This equation describes the dynamics of magnetization, with a non-convexity constraint of unit length of the magnetization. The numerical method is based on the second-order backward differentiation formula in time, combined with an implicit treatment of the linear diffusion term and explicit extrapolation for the nonlinear terms. Afterward, a projection step is applied to normalize the numerical solution at a point-wise level. This numerical scheme has shown extensive advantages in the practical computations for the physical model with large damping parameters, which comes from the fact that only a linear system with constant coefficients (independent of both time and the updated magnetization) needs to be solved at each time step, and has greatly improved the numerical efficiency. Meanwhile, a theoretical analysis for this linear numerical scheme has not been available. In this paper, we provide a rigorous error estimate of the numerical scheme, in the discrete ℓ ∞ (0, T ; ℓ 2 ) ∩ ℓ 2 (0, T ; H 1 h ) norm, under suitable regularity assumptions and reasonable ratio between the time step-size and the spatial mesh-size. In particular, the projection operation is nonlinear, and a stability estimate for the projection step turns out to be highly challenging. Such a stability estimate is derived in details, which will play an essential role in the convergence analysis for the numerical scheme, if the damping parameter is greater than 3.

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Section: 2mentioning

confidence: 99%

Convergence Analysis of A Second-order Accurate, Linear Numerical Scheme for The Landau-Lifshitz Equation with Large Damping Parameters

Cai¹,

Chen²,

Wang³

et al. 2021

Preprint

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“…For micromagnetics simulations, to update m n+1 , only h n s is needed. Following [17], we establish the unique solvability of the proposed scheme (16) as follows.…”

Section: A Second-order Semi-implicit Finite Difference Schemementioning

confidence: 99%

“…Second, in (16), a projection step is applied after solving (17). The following proposition guarantees that the denominator is always nonzero.…”

Section: A Second-order Semi-implicit Finite Difference Schemementioning

confidence: 99%

“…First order semi-implicit schemes include the Gauss-Seidel projection method [14,15] and the semi-implicit backward Euler method [16]. The second order semi-implicit projection method with backward differentiation formula has been recently proposed and its second-order accuracy is established theoretically [17,18]. Closely related to the current work, the implicit midpoint scheme is available in the literature [19].…”

Section: Introductionmentioning

confidence: 99%

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A second-order semi-implicit method for the inertial Landau-Lifshitz-Gilbert equation

Li¹,

Yang

Lan

et al. 2021

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Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization. Its dynamics spans over a wide range of timescales from femtosecond to picosecond, and then to nanosecond. The Landau-Lifshitz-Gilbert (LLG) equation has widely been used in micromagnetics simulations over decades. Recent theoretical and experimental advances show that the inertia of magnetization emerges at sub-picoseconds and contributes to the ultrafast magnetization dynamics which cannot be captured intrinsically by the LLG equation. Therefore, as a generalization, the inertial LLG (iLLG) equation is proposed to model the ultrafast magnetization dynamics. Mathematically, the LLG equation is a nonlinear system of parabolic type with (possible) degeneracy. However, the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy, and exhibits more complicated structures. It behaves like a hyperbolic system at the sub-picosecond scale while behaves like a parabolic system at larger timescales. Such hybrid behaviors impose additional difficulties on designing numerical methods for the iLLG equation. In this work, we propose a second-order semi-implicit scheme to solve the iLLG equation. The second temporal derivative of magnetization is approximated by the standard centered difference scheme and the first derivative is approximated by the midpoint scheme involving three time steps. The nonlinear terms are treated semi-implicitly using one-sided interpolation with the second-order accuracy. At each step, the unconditionally unique solvability of the unsymmetric linear system of equations in the proposed method is proved with a detailed discussion on the condition number. Numerically, the second-order accuracy in both time and space is verified. Using the proposed method, the inertial effect of ferromagnetics is observed in micromagnetics simulations at small timescales, in consistency with the hyperbolic property of the model at sub-picoseconds. For long time simulations, the results of the iLLG model are in nice agreements with those of the LLG model, in consistency with the parabolic feature of the iLLG model at larger timescales.

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“…Therefore, we will employ the improved GSPM (referred to Scheme A in [17]) in the following parts. In [22], the semi-implicit schemes are constructed by using the backward differentiation formula (BDF) and one-sided interpolation. The second-order BDF scheme (BDF2) is proven to converge with the second-order accuracy in both space and time [5].…”

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confidence: 99%

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

et al. 2022

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<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ 60\% $\end{document}</tex-math></inline-formula> compared with the state-of-the-art of GSPM for micromagnetics simulations. For Standard Problems #4 and #5 from National Institute of Standards and Technology, GSPM-BDF2 reduces the computational time over the popular software OOMMF by <inline-formula><tex-math id="M2">\begin{document}$ 82\% $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ 96\% $\end{document}</tex-math></inline-formula>, respectively. Thus, the proposed method provides a more efficient choice for micromagnetics simulations.</p>

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Second-order semi-implicit projection methods for micromagnetics simulations

Cited by 26 publications

References 39 publications

Convergence Analysis of A Second-order Accurate, Linear Numerical Scheme for The Landau-Lifshitz Equation with Large Damping Parameters

Convergence Analysis of A Second-order Accurate, Linear Numerical Scheme for The Landau-Lifshitz Equation with Large Damping Parameters

A second-order semi-implicit method for the inertial Landau-Lifshitz-Gilbert equation

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

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