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 & 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

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

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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

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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]

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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

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

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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 $p_{\xi}^{n}-p_{\xi}^{0}$
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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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