American Institute of Mathematical Sciences

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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  March 2018 Early access  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 and 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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

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 $p_{\xi}^{n}-p_{\xi}^{0}$
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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
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Figure 3.  Convergence rates of numerical solutions by the methods $GS_{h}^{2}$, $\tilde{G}_{h}^{2}$, $GS_{h}^{4}$ and $G_{h}^{4}$
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Figure 4.  Left: The errors of the energy. Right: The errors of the angular momentum
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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 $[0, 10^{5}h]$
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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 $[0, 5\times10^{5}h]$
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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 $[0, 5\times10^{5}h]$
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