American Institute of Mathematical Sciences

Advanced
March  2018, 23(2): 573-585. doi: 10.3934/dcdsb.2018034

High order Gauss-Seidel schemes for charged particle dynamics

Yuezheng Gong 1, , Jiaquan Gao 2, and  Yushun Wang 3,,

1. 

College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2. 

School of Computer Science and Technology, Nanjing Normal University, Nanjing 210023, China
3. 

Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

* Corresponding author:wangyushun@njnu.edu.cn(Y. Wang)

Received  October 2015 Revised  July 2017 Published  December 2017

Gauss-Seidel projection methods are designed for achieving desirable long-term computational efficiency and reliability in micromagnetics simulations. While conventional Gauss-Seidel schemes are explicit, easy to use and furnish a better stability as compared to Euler's method, their order of accuracy is only one. This paper proposes an improved Gauss-Seidel methodology for particle simulations of magnetized plasmas. A novel new class of high order schemes are implemented via composition strategies. The new algorithms acquired are not only explicit and symmetric, but also volume-preserving together with their adjoint schemes. They are highly favorable for long-term computations. The new high order schemes are then utilized for simulating charged particle motions under the Lorentz force. Our experiments indicate a remarkable satisfaction of the energy preservation and angular momentum conservation of the numerical methods in multi-scale plasma dynamics computations.

Keywords: Gauss-Seidel scheme, high order accuracy, long-time computation, volume-preserving, conservations.
Mathematics Subject Classification: 60E10, 60J10, 60J27, 60J35.
Citation: Yuezheng Gong, Jiaquan Gao, Yushun Wang. High order Gauss-Seidel schemes for charged particle dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 573-585. doi: 10.3934/dcdsb.2018034
References:
[1]

P. M. Bellan, Fundamentals of Plasma Physics, 1$^{st}$ edition, Cambridge University Press, 2008. doi: 10.1017/CBO9780511807183.  Google Scholar

[2]

C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, Ser. Plasma Phys. Taylor & Francis, 2005. Google Scholar

[3]

J. Boris, Proceedings of the fourth conference on numerical simulation of plasmas, Washington D.C., (1970), 3-67.   Google Scholar

[4]

S. A. Chin, Symplectic and energy-conserving algorithms for solving magnetic field trajectories, Phy. Rev. E, 77 (2008), 066401, 12pp.  Google Scholar

[5]

W. N. E and X. P. Wang, Numerical methods for the Landau-Lifshitz equation, SIAM J. Numer. Anal., 38 (2000), 1647-1665.  doi: 10.1137/S0036142999352199.  Google Scholar

[6]

K. Feng, Symplectic, contact and volume-preserving algorithms, in: Z. C. Shi, T. Ushijima (Eds. ), Proc. 1st China-Japan Conf. on Computation of Differential Equations and Dynamical Systems, World Scientific, Singapore, 1993, 1–28. Google Scholar

[7]

K. Feng and Z. Shang, Volume-preserving algorithms for source-free dynamical systems, Numer. Math., 71 (1995), 451-463.  doi: 10.1007/s002110050153.  Google Scholar

[8]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2006.  Google Scholar

[9]

Y. He, Y. J. Sun, J. Liu and H. Qin, Volume-preserving algorithms for charged particle dynamics, J. Comput. Phys., 281 (2015), 135-147.  doi: 10.1016/j.jcp.2014.10.032.  Google Scholar

[10]

Y. He, Z. Zhou, Y. Sun, J. Liu and H. Qin, Explicit $K$-symplectic algorithms for charged particle dynamics, Phys. Lett. A, 381 (2017), 568-573.  doi: 10.1016/j.physleta.2016.12.031.  Google Scholar

[11]

R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Phys. Fluids, 24 (1981), 1730-1749.  doi: 10.1063/1.863594.  Google Scholar

[12]

R. G. Littlejohn, Variational principles of guiding centre motion, J. Plasma Phys., 29 (1983), 111-125.  doi: 10.1017/S002237780000060X.  Google Scholar

[13]

B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge: Cambridge University Press, 2004.  Google Scholar

[14]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, Berlin, 1999.  Google Scholar

[15]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar

[16]

H. Qin, S. X. Zhang, J. Y. Xiao, J. Liu, Y. J. Sun and W. M. Tang, Why is Boris algorithm so good?, Phys. Plasmas, 20 (2013), 084503. doi: 10.1063/1.4818428.  Google Scholar

[17]

H. Qin and X. Guan, Variational symplectic integrator for long-time simulations of the guiding-center motion of charged particles in general magnetic fields, Phys. Rev. Lett. , 100 (2008), 035006. doi: 10.1103/PhysRevLett.100.035006.  Google Scholar

[18]

Z. Shang, Construction of volume-preserving difference schemes for source-free systems via generating functions, J. Comput. Math., 12 (1994), 265-272.   Google Scholar

[19]

P. H. Stoltz, J. R. Cary, G. Penn and J. Wurtele, Efficiency of a Boris like integration scheme with spatial stepping, Phys. Rev. Spec. Top. Accel. Beams, 5 (2002), 094001. doi: 10.1103/PhysRevSTAB.5.094001.  Google Scholar

[20]

Y. Sun, A class of volume-preserving numerical algorithms, Appl. Math. Comput., 206 (2008), 841-852.  doi: 10.1016/j.amc.2008.10.004.  Google Scholar

[21]

X. P. Wang, C. J. Garcia-Cervera and W. N. E, A Gauss-Seidel Projection Method for Micromagnetics Simulations, J. Comput. Phys., 171 (2001), 357-372.  doi: 10.1006/jcph.2001.6793.  Google Scholar

[22]

S. D. Webb, Symplectic intergration of magnetic systems, J. Comput. Phys., 270 (2014), 570-576.  doi: 10.1016/j.jcp.2014.03.049.  Google Scholar

[23]

Y. K. Wu, E. Forest and D. S. Robin, Explicit symplectic integrator for s-dependent static magnetic field, Phys. Rev. E, 68 (2003), 046502. doi: 10.1103/PhysRevE.68.046502.  Google Scholar

[24]

S. X. Zhang, Y. S. Jia and Q. Z. Sun, Comment on "Symplectic integration of magnetic systems" by Stephen D. Webb [J. Comput. Phys. 270 (2014) 570-576], J. Comput. Phys., 282 (2015), 43-46.  doi: 10.1016/j.jcp.2014.10.062.  Google Scholar

show all references

References:
[1]

P. M. Bellan, Fundamentals of Plasma Physics, 1$^{st}$ edition, Cambridge University Press, 2008. doi: 10.1017/CBO9780511807183.  Google Scholar

[2]

C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, Ser. Plasma Phys. Taylor & Francis, 2005. Google Scholar

[3]

J. Boris, Proceedings of the fourth conference on numerical simulation of plasmas, Washington D.C., (1970), 3-67.   Google Scholar

[4]

S. A. Chin, Symplectic and energy-conserving algorithms for solving magnetic field trajectories, Phy. Rev. E, 77 (2008), 066401, 12pp.  Google Scholar

[5]

W. N. E and X. P. Wang, Numerical methods for the Landau-Lifshitz equation, SIAM J. Numer. Anal., 38 (2000), 1647-1665.  doi: 10.1137/S0036142999352199.  Google Scholar

[6]

K. Feng, Symplectic, contact and volume-preserving algorithms, in: Z. C. Shi, T. Ushijima (Eds. ), Proc. 1st China-Japan Conf. on Computation of Differential Equations and Dynamical Systems, World Scientific, Singapore, 1993, 1–28. Google Scholar

[7]

K. Feng and Z. Shang, Volume-preserving algorithms for source-free dynamical systems, Numer. Math., 71 (1995), 451-463.  doi: 10.1007/s002110050153.  Google Scholar

[8]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2006.  Google Scholar

[9]

Y. He, Y. J. Sun, J. Liu and H. Qin, Volume-preserving algorithms for charged particle dynamics, J. Comput. Phys., 281 (2015), 135-147.  doi: 10.1016/j.jcp.2014.10.032.  Google Scholar

[10]

Y. He, Z. Zhou, Y. Sun, J. Liu and H. Qin, Explicit $K$-symplectic algorithms for charged particle dynamics, Phys. Lett. A, 381 (2017), 568-573.  doi: 10.1016/j.physleta.2016.12.031.  Google Scholar

[11]

R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Phys. Fluids, 24 (1981), 1730-1749.  doi: 10.1063/1.863594.  Google Scholar

[12]

R. G. Littlejohn, Variational principles of guiding centre motion, J. Plasma Phys., 29 (1983), 111-125.  doi: 10.1017/S002237780000060X.  Google Scholar

[13]

B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge: Cambridge University Press, 2004.  Google Scholar

[14]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, Berlin, 1999.  Google Scholar

[15]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar

[16]

H. Qin, S. X. Zhang, J. Y. Xiao, J. Liu, Y. J. Sun and W. M. Tang, Why is Boris algorithm so good?, Phys. Plasmas, 20 (2013), 084503. doi: 10.1063/1.4818428.  Google Scholar

[17]

H. Qin and X. Guan, Variational symplectic integrator for long-time simulations of the guiding-center motion of charged particles in general magnetic fields, Phys. Rev. Lett. , 100 (2008), 035006. doi: 10.1103/PhysRevLett.100.035006.  Google Scholar

[18]

Z. Shang, Construction of volume-preserving difference schemes for source-free systems via generating functions, J. Comput. Math., 12 (1994), 265-272.   Google Scholar

[19]

P. H. Stoltz, J. R. Cary, G. Penn and J. Wurtele, Efficiency of a Boris like integration scheme with spatial stepping, Phys. Rev. Spec. Top. Accel. Beams, 5 (2002), 094001. doi: 10.1103/PhysRevSTAB.5.094001.  Google Scholar

[20]

Y. Sun, A class of volume-preserving numerical algorithms, Appl. Math. Comput., 206 (2008), 841-852.  doi: 10.1016/j.amc.2008.10.004.  Google Scholar

[21]

X. P. Wang, C. J. Garcia-Cervera and W. N. E, A Gauss-Seidel Projection Method for Micromagnetics Simulations, J. Comput. Phys., 171 (2001), 357-372.  doi: 10.1006/jcph.2001.6793.  Google Scholar

[22]

S. D. Webb, Symplectic intergration of magnetic systems, J. Comput. Phys., 270 (2014), 570-576.  doi: 10.1016/j.jcp.2014.03.049.  Google Scholar

[23]

Y. K. Wu, E. Forest and D. S. Robin, Explicit symplectic integrator for s-dependent static magnetic field, Phys. Rev. E, 68 (2003), 046502. doi: 10.1103/PhysRevE.68.046502.  Google Scholar

[24]

S. X. Zhang, Y. S. Jia and Q. Z. Sun, Comment on "Symplectic integration of magnetic systems" by Stephen D. Webb [J. Comput. Phys. 270 (2014) 570-576], J. Comput. Phys., 282 (2015), 43-46.  doi: 10.1016/j.jcp.2014.10.062.  Google Scholar

Figure 1.  The fourth order explicit method $RK4$ is applied to the simple 2D dynamics with step size $h = \pi/10$. (a): The orbit in the first $2691$ steps. (b): The orbit after $2.7\times10^{5}$ steps. (c): Energy error $H^{n}-H^{0}$. (d): Angular momentum error $p_{\xi}^{n}-p_{\xi}^{0}$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Numerical orbits of the symmetric and volume-preserving methods with time step $h = \pi/10$. (a): The orbit after $5\times10^{5}$ steps by the second order method. (b): The orbit after $2.5\times10^{5}$ steps by the fourth order method
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Convergence rates of numerical solutions by the methods $GS_{h}^{2}$, $\tilde{G}_{h}^{2}$, $GS_{h}^{4}$ and $G_{h}^{4}$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Left: The errors of the energy. Right: The errors of the angular momentum
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Relative errors of the energy $H$ and the angular momentum $p_{\xi}$ as a function of time $t\equiv nh$. The step size is $h = \pi/10$, and the integration time interval is $[0, 10^{5}h]$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Numerical orbits. (a): Banana orbit by the $RK4$. (b): Transit orbit by the $RK4$. (c): Banana orbit by the volume-preserving methods. (d): Transit orbit by the volume-preserving methods. The step size is $h = \pi/10$, and the integration time interval is $[0, 5\times10^{5}h]$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Relative errors of the energy $H$ and the angular momentum $p_{\xi}$ as a function of time $t\equiv nh$. The step size is $h = \pi/10$, and the integration time interval is $[0, 5\times10^{5}h]$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Rhudaina Z. Mohammad, Karel Švadlenka. Multiphase volume-preserving interface motions via localized signed distance vector scheme. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 969-988. doi: 10.3934/dcdss.2015.8.969
[2]

H. E. Lomelí, J. D. Meiss. Generating forms for exact volume-preserving maps. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 361-377. doi: 10.3934/dcdss.2009.2.361
[3]

Horst Osberger. Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 405-434. doi: 10.3934/dcds.2017017
[4]

Ning Zhang. A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems. Journal of Industrial & Management Optimization, 2020, 16 (2) : 991-1008. doi: 10.3934/jimo.2018189
[5]

Fuzhong Cong, Hongtian Li. Quasi-effective stability for a nearly integrable volume-preserving mapping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1959-1970. doi: 10.3934/dcdsb.2015.20.1959
[6]

Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146
[7]

Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283
[8]

Zeyu Xia, Xiaofeng Yang. A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020089
[9]

Vadim Kaloshin, Maria Saprykina. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 611-640. doi: 10.3934/dcds.2006.15.611
[10]

Joackim Bernier, Michel Mehrenberger. Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium. Kinetic & Related Models, 2020, 13 (1) : 129-168. doi: 10.3934/krm.2020005
[11]

Manuel Núñez. The long-time evolution of mean field magnetohydrodynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 465-478. doi: 10.3934/dcdsb.2004.4.465
[12]

Florian Schneider, Jochen Kall, Graham Alldredge. A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry. Kinetic & Related Models, 2016, 9 (1) : 193-215. doi: 10.3934/krm.2016.9.193
[13]

Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri. Long-time behavior of solutions of a BBM equation with generalized damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1897-1915. doi: 10.3934/dcdsb.2015.20.1897
[14]

A. Kh. Khanmamedov. Long-time behaviour of doubly nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1373-1400. doi: 10.3934/cpaa.2009.8.1373
[15]

Marcio Antonio Jorge da Silva, Vando Narciso. Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*. Evolution Equations & Control Theory, 2017, 6 (3) : 437-470. doi: 10.3934/eect.2017023
[16]

Igor Chueshov, Stanislav Kolbasin. Long-time dynamics in plate models with strong nonlinear damping. Communications on Pure & Applied Analysis, 2012, 11 (2) : 659-674. doi: 10.3934/cpaa.2012.11.659
[17]

Yihong Du, Yoshio Yamada. On the long-time limit of positive solutions to the degenerate logistic equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 123-132. doi: 10.3934/dcds.2009.25.123
[18]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[19]

Min Chen, Olivier Goubet. Long-time asymptotic behavior of dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 509-528. doi: 10.3934/dcds.2007.17.509
[20]

Yuguo Lin, Daqing Jiang. Long-time behaviour of a perturbed SIR model by white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1873-1887. doi: 10.3934/dcdsb.2013.18.1873

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (92)
  • HTML views (169)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences