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doi: 10.3934/dcdsb.2022002
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A Gauss-Seidel projection method with the minimal number of updates for the stray field in micromagnetics simulations

Panchi Li 1, , Zetao Ma 1, , Rui Du 1,2,, and  Jingrun Chen 1,2,3,4,,

1. 

School of Mathematical Sciences, Soochow University, Suzhou, 215006, China
2. 

Mathematical Center for Interdisciplinary Research, Soochow University, Suzhou, 215006, China
3. 

School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
4. 

Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou, Jiangsu 215123, China

*Corresponding authors: Rui Du and Jingrun Chen

Received  June 2021 Revised  September 2021 Early access January 2022

Fund Project: 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.

Keywords: Micromagnetics simulation, Landau-Lifshitz equation, Gauss-Seidel projection method, backward differentiation formula, Stray field.
Mathematics Subject Classification: Primary: 35Q99, 65Z05, 65M06.
Citation: Panchi Li, Zetao Ma, Rui Du, Jingrun Chen. A Gauss-Seidel projection method with the minimal number of updates for the stray field in micromagnetics simulations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022002
References:
[1]

C. Abert, L. Exl, G. Selke, A. Drews and T. Schrefl, Numerical methods for the stray-field calculation: A comparison of recently developed algorithms, Journal of Magnetism and Magnetic Materials, 326 (2013), 176-185.  doi: 10.1016/j.jmmm.2012.08.041.

[2]

S. Bartels and A. Prohl, Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 44 (2006), 1405-1419.  doi: 10.1137/050631070.

[3]

F. Bruckner, A. Ducevic, P. Heistracher, C. Abert and D. Suess, Strayfield calculation for micromagnetic simulations using true periodic boundary conditions, Sci. Rep., 11 (2021), 9202.  doi: 10.1038/s41598-021-88541-9.

[4]

J. R. Cash and A. H. Karp, A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Trans. Math. Soft., 16 (1990), 201-222.  doi: 10.1145/79505.79507.

[5]

J. Chen, C. Wang and C. Xie, Convergence analysis of a second-order semi-implicit projection method for Landau-Lifshitz equation, Appl. Numer. Math., 168 (2021), 55-74.  doi: 10.1016/j.apnum.2021.05.027.

[6]

I. Cimrák, Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation with an exchange field, IMA J. Numer. Anal., 25 (2005), 611-634.  doi: 10.1093/imanum/dri011.

[7]

F. G. Di, P. Carl-Martin, P. Dirk, R. Michele and S. Bernhard, Linear second-order IMEX-type integrator for the (eddy current) Landau-Lifshitz-Gilbert equation, IMA J. Numer. Anal., 40 (2019), 2802-2838.  doi: 10.1093/imanum/drz046.

[8]

M. J. Donahue and D. G. Porter, OOMMF User's Guide, 2019. http://math.nist.gov/oommf/.

[9]

H. Fangohr, T. Fischbacher, M. Franchin, G. Bordignon, J. Generowicz, A. Knittel, M. Walter and M. Albert, NMAG User Manual (0.2.1), 2012.

[10]

A. Fuwa, T. Ishiwata and M. Tsutsumi, Finite difference scheme for the Landau-Lifshitz equation, Jpn. J. Ind. Appl. Math., 29 (2012), 83-110.  doi: 10.1007/s13160-011-0054-9.

[11]

C. J. García-Cervera, Numerical micromagnetics: A review, Bol. Soc. Esp. Mat. Apl., 39 (2007), 103-135. 

[12]

C. J. García-Cervera and We inan E, Improved Gauss-Seidel projection method for micromagnetics simulations, IEEE Trans. Magn., 39 (2003), 1766-1770. 

[13]

T. L. Gilbert, A Lagrangian formulation of gyromagnetic equation of the magnetization field, Phys. Rev., 100 (1955), 1243-1255. 

[14]

D. Jeong and J. Kim, A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping, J. Comput. Appl. Math., 234 (2010), 613-623.  doi: 10.1016/j.cam.2010.01.002.

[15]

L. D. Landau and E. M. Lifshitz, On the theory of the dispersion of magetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion, 8 (1935), 153-169. 

[16]

P. Li, J. Chen, R. Du and X.-P. Wang, Numerical methods for antiferrimagnets, IEEE Trans. Magn., 56 (2020), 7200509. 

[17]

P. Li, C. Xie, R. Du, J. Chen and X.-P. Wang, Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation, J. Comput. Phys., 401 (2020), 109046, 12 pp. doi: 10.1016/j.jcp.2019.109046.

[18]

M. Najafi, B. Krüger, S. Bohlens, M. Franchin, H. Fangohr, A. Vanhaverbeke, R. Allenspach, M. Bolte, U. Merkt, D. Pfannkuche, D. P. F. Möller and G. Meier, Proposal for a standard problem for micromagnetic simulations including spin-transfer torque, J. Appl. Phys., 105 (2009), 113914.  doi: 10.1063/1.3126702.

[19]

D. Praetorius, M. Ruggeri and B. Stiftner, Convergence of an implicit-explicit midpoint scheme for computational micromagnetics, Comput. Math. Appl., 75 (2018), 1719-1738.  doi: 10.1016/j.camwa.2017.11.028.

[20]

A. Romeo, G. Finocchio, M. Carpentieri, L. Torres, G. Consolo and B. Azzerboni, A numerical solution of the magnetization reversal modeling in a permalloy thin film using fifth order Runge-Kutta method with adaptive step size control, Phys. B Condens. Matter, 403 (2008), 464-468.  doi: 10.1016/j.physb.2007.08.076.

[21]

X.-P. Wang, C. J. García-Cervera and W. E, A Gauss-Seidel projection method for micromagnetics simulations, J. Comput. Phys., 171 (2001), 357-372.  doi: 10.1006/jcph.2001.6793.

[22]

C. Xie, C. J. García-Cervera, C. Wang, Z. Zhou and J. Chen, Second-order semi-implicit projection methods for micromagnetics simulations, J. Comput. Phys., 404 (2020), 109104, 14 pp. doi: 10.1016/j.jcp.2019.109104.

[23]

H. Yamada and N. Hayashi, Implicit solution of the Landau-Lifshitz-Gilbert equation by the Crank-Nicolson method, J. Magn. Soc. Jpn., 28 (2004), 924-931. 

[24]

L. Yang, Current Induced Domain Wall Motion: Analysis and Simulation, Ph. D thesis, HKUST, 2008.

[25]

L. Yang, J. Chen and G. Hu, A framework of the finite element solution of the Landau-Lifshitz-Gilbert equation on tetrahedral meshes, J. Comput. Phys., 431 (2021), Paper No. 110142, 17 pp. doi: 10.1016/j.jcp.2021.110142.

[26]

L. Yang and G. Hu, An adaptive finite element solver for demagnetization field calculation, Adv. Appl. Math. Mech, 11 (2019), 1048-1063.  doi: 10.4208/aamm.OA-2018-0236.

[27]

S. Zhang and Z. Li, Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets, Phys. Rev. Lett., 93 (2004), 127204.  doi: 10.1103/PhysRevLett.93.127204.

[28]

Micromagnetic Modeling Activity Group, National Institute of Standards and Technology, 2020. https://www.ctcms.nist.gov/rdm/mumag.org.html.

[29]

I. Žutić, J. Fabian and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys., 76 (2004), 323-410. 

show all references

References:
[1]

C. Abert, L. Exl, G. Selke, A. Drews and T. Schrefl, Numerical methods for the stray-field calculation: A comparison of recently developed algorithms, Journal of Magnetism and Magnetic Materials, 326 (2013), 176-185.  doi: 10.1016/j.jmmm.2012.08.041.

[2]

S. Bartels and A. Prohl, Convergence of an implicit finite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 44 (2006), 1405-1419.  doi: 10.1137/050631070.

[3]

F. Bruckner, A. Ducevic, P. Heistracher, C. Abert and D. Suess, Strayfield calculation for micromagnetic simulations using true periodic boundary conditions, Sci. Rep., 11 (2021), 9202.  doi: 10.1038/s41598-021-88541-9.

[4]

J. R. Cash and A. H. Karp, A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Trans. Math. Soft., 16 (1990), 201-222.  doi: 10.1145/79505.79507.

[5]

J. Chen, C. Wang and C. Xie, Convergence analysis of a second-order semi-implicit projection method for Landau-Lifshitz equation, Appl. Numer. Math., 168 (2021), 55-74.  doi: 10.1016/j.apnum.2021.05.027.

[6]

I. Cimrák, Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation with an exchange field, IMA J. Numer. Anal., 25 (2005), 611-634.  doi: 10.1093/imanum/dri011.

[7]

F. G. Di, P. Carl-Martin, P. Dirk, R. Michele and S. Bernhard, Linear second-order IMEX-type integrator for the (eddy current) Landau-Lifshitz-Gilbert equation, IMA J. Numer. Anal., 40 (2019), 2802-2838.  doi: 10.1093/imanum/drz046.

[8]

M. J. Donahue and D. G. Porter, OOMMF User's Guide, 2019. http://math.nist.gov/oommf/.

[9]

H. Fangohr, T. Fischbacher, M. Franchin, G. Bordignon, J. Generowicz, A. Knittel, M. Walter and M. Albert, NMAG User Manual (0.2.1), 2012.

[10]

A. Fuwa, T. Ishiwata and M. Tsutsumi, Finite difference scheme for the Landau-Lifshitz equation, Jpn. J. Ind. Appl. Math., 29 (2012), 83-110.  doi: 10.1007/s13160-011-0054-9.

[11]

C. J. García-Cervera, Numerical micromagnetics: A review, Bol. Soc. Esp. Mat. Apl., 39 (2007), 103-135. 

[12]

C. J. García-Cervera and We inan E, Improved Gauss-Seidel projection method for micromagnetics simulations, IEEE Trans. Magn., 39 (2003), 1766-1770. 

[13]

T. L. Gilbert, A Lagrangian formulation of gyromagnetic equation of the magnetization field, Phys. Rev., 100 (1955), 1243-1255. 

[14]

D. Jeong and J. Kim, A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping, J. Comput. Appl. Math., 234 (2010), 613-623.  doi: 10.1016/j.cam.2010.01.002.

[15]

L. D. Landau and E. M. Lifshitz, On the theory of the dispersion of magetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion, 8 (1935), 153-169. 

[16]

P. Li, J. Chen, R. Du and X.-P. Wang, Numerical methods for antiferrimagnets, IEEE Trans. Magn., 56 (2020), 7200509. 

[17]

P. Li, C. Xie, R. Du, J. Chen and X.-P. Wang, Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation, J. Comput. Phys., 401 (2020), 109046, 12 pp. doi: 10.1016/j.jcp.2019.109046.

[18]

M. Najafi, B. Krüger, S. Bohlens, M. Franchin, H. Fangohr, A. Vanhaverbeke, R. Allenspach, M. Bolte, U. Merkt, D. Pfannkuche, D. P. F. Möller and G. Meier, Proposal for a standard problem for micromagnetic simulations including spin-transfer torque, J. Appl. Phys., 105 (2009), 113914.  doi: 10.1063/1.3126702.

[19]

D. Praetorius, M. Ruggeri and B. Stiftner, Convergence of an implicit-explicit midpoint scheme for computational micromagnetics, Comput. Math. Appl., 75 (2018), 1719-1738.  doi: 10.1016/j.camwa.2017.11.028.

[20]

A. Romeo, G. Finocchio, M. Carpentieri, L. Torres, G. Consolo and B. Azzerboni, A numerical solution of the magnetization reversal modeling in a permalloy thin film using fifth order Runge-Kutta method with adaptive step size control, Phys. B Condens. Matter, 403 (2008), 464-468.  doi: 10.1016/j.physb.2007.08.076.

[21]

X.-P. Wang, C. J. García-Cervera and W. E, A Gauss-Seidel projection method for micromagnetics simulations, J. Comput. Phys., 171 (2001), 357-372.  doi: 10.1006/jcph.2001.6793.

[22]

C. Xie, C. J. García-Cervera, C. Wang, Z. Zhou and J. Chen, Second-order semi-implicit projection methods for micromagnetics simulations, J. Comput. Phys., 404 (2020), 109104, 14 pp. doi: 10.1016/j.jcp.2019.109104.

[23]

H. Yamada and N. Hayashi, Implicit solution of the Landau-Lifshitz-Gilbert equation by the Crank-Nicolson method, J. Magn. Soc. Jpn., 28 (2004), 924-931. 

[24]

L. Yang, Current Induced Domain Wall Motion: Analysis and Simulation, Ph. D thesis, HKUST, 2008.

[25]

L. Yang, J. Chen and G. Hu, A framework of the finite element solution of the Landau-Lifshitz-Gilbert equation on tetrahedral meshes, J. Comput. Phys., 431 (2021), Paper No. 110142, 17 pp. doi: 10.1016/j.jcp.2021.110142.

[26]

L. Yang and G. Hu, An adaptive finite element solver for demagnetization field calculation, Adv. Appl. Math. Mech, 11 (2019), 1048-1063.  doi: 10.4208/aamm.OA-2018-0236.

[27]

S. Zhang and Z. Li, Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets, Phys. Rev. Lett., 93 (2004), 127204.  doi: 10.1103/PhysRevLett.93.127204.

[28]

Micromagnetic Modeling Activity Group, National Institute of Standards and Technology, 2020. https://www.ctcms.nist.gov/rdm/mumag.org.html.

[29]

I. Žutić, J. Fabian and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys., 76 (2004), 323-410. 

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
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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 $
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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
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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
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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
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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} $
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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 $)
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#(linear systems of equations) (dofs) #(stray field updates) (dofs)
GSPM 5 ($ 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
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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
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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%
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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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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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