American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production
  • DCDS-B Home
  • This Issue
  • Next Article
    Synchronization of dynamical systems on Riemannian manifolds by an extended PID-type control theory: Numerical evaluation
doi: 10.3934/dcdsb.2022002
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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} $
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download as excel

#(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 Options

Download as excel

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 Options

Download as excel

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 Options

Download as excel

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%
Table Options

Download as excel

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%
[1]

Xin Yang, Nan Wang, Lingling Xu. A parallel Gauss-Seidel method for convex problems with separable structure. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 557-570. doi: 10.3934/naco.2020051
[2]

Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644
[3]

Ning Zhang. A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems. Journal of Industrial and Management Optimization, 2020, 16 (2) : 991-1008. doi: 10.3934/jimo.2018189
[4]

Yuezheng Gong, Jiaquan Gao, Yushun Wang. High order Gauss-Seidel schemes for charged particle dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 573-585. doi: 10.3934/dcdsb.2018034
[5]

Xueke Pu, Boling Guo, Jingjun Zhang. Global weak solutions to the 1-D fractional Landau-Lifshitz equation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 199-207. doi: 10.3934/dcdsb.2010.14.199
[6]

Wei Deng, Baisheng Yan. On Landau-Lifshitz equations of no-exchange energy models in ferromagnetics. Evolution Equations and Control Theory, 2013, 2 (4) : 599-620. doi: 10.3934/eect.2013.2.599
[7]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure and Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
[8]

Ze Li, Lifeng Zhao. Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 607-638. doi: 10.3934/dcds.2019025
[9]

Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071
[10]

Catherine Choquet, Mohammed Moumni, Mouhcine Tilioua. Homogenization of the Landau-Lifshitz-Gilbert equation in a contrasted composite medium. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 35-57. doi: 10.3934/dcdss.2018003
[11]

Boling Guo, Fangfang Li. Global smooth solution for the Sipn-Polarized transport equation with Landau-Lifshitz-Bloch equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2825-2840. doi: 10.3934/dcdsb.2020034
[12]

Tram Thi Ngoc Nguyen, Anne Wald. On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging. Inverse Problems and Imaging, 2022, 16 (1) : 89-117. doi: 10.3934/ipi.2021042
[13]

Jing Li, Boling Guo, Lan Zeng, Yitong Pei. Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1345-1360. doi: 10.3934/dcdsb.2019230
[14]

Z. B. Ibrahim, N. A. A. Mohd Nasir, K. I. Othman, N. Zainuddin. Adaptive order of block backward differentiation formulas for stiff ODEs. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 95-106. doi: 10.3934/naco.2017006
[15]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867
[16]

Ke Chen, Yiqiu Dong, Michael Hintermüller. A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration. Inverse Problems and Imaging, 2011, 5 (2) : 323-339. doi: 10.3934/ipi.2011.5.323
[17]

Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133
[18]

Rashad M. Asharabi, Jürgen Prestin. Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4143-4158. doi: 10.3934/cpaa.2020185
[19]

Andrei Agrachev, Ugo Boscain, Mario Sigalotti. A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 801-822. doi: 10.3934/dcds.2008.20.801
[20]

Kolade M. Owolabi, Edson Pindza. Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 835-851. doi: 10.3934/dcdss.2020048

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (275)
  • HTML views (153)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy