# American Institute of Mathematical Sciences

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

## High order Gauss-Seidel schemes for charged particle dynamics

 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.

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

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