# American Institute of Mathematical Sciences

December  2019, 6(2): 429-448. doi: 10.3934/jcd.2019022

## Study of adaptive symplectic methods for simulating charged particle dynamics

 1 LSEC, ICMSEC, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China 2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3 Department of Engineering and Applied Physics, USTC, Hefei, Anhui 230026, China 4 Key Laboratory of Geospace Environment, CAS, Hefei, Anhui 230026, China

* Corresponding author: Yajuan Sun

Received  March 2019 Revised  September 2019 Published  November 2019

In plasma simulations, numerical methods with high computational efficiency and long-term stability are needed. In this paper, symplectic methods with adaptive time steps are constructed for simulating the dynamics of charged particles under the electromagnetic field. With specifically designed step size functions, the motion of charged particles confined in a Penning trap under three different magnetic fields is studied, and also the dynamics of runaway electrons in tokamaks is investigated. The numerical experiments are performed to show the efficiency of the new derived adaptive symplectic methods.

Citation: Yanyan Shi, Yajuan Sun, Yulei Wang, Jian Liu. Study of adaptive symplectic methods for simulating charged particle dynamics. Journal of Computational Dynamics, 2019, 6 (2) : 429-448. doi: 10.3934/jcd.2019022
##### References:
 [1] G. Benettin and P. Sempio, Adiabatic invariants and trapping of a point charge in a strong nonuniform magnetic field, Nonlinearity, 7 (1994), 281-303.  doi: 10.1088/0951-7715/7/1/014.  Google Scholar [2] M. P. Calvo and J. M. Sanz-Serna, The Development of variable-step symplectic integrators, with application to the two-body problem, SIAM Journal on Scientific Computing, 14 (1993), 936-952.  doi: 10.1137/0914057.  Google Scholar [3] H. Dreicer, Electron and ion runaway in a fully ionized gas, Physical Review, 115 (1959), 238-249.  doi: 10.1103/PhysRev.115.238.  Google Scholar [4] B. Gladman, M. Duncan and J. Candy, Symplectic integrators for long-term integrations in celestial mechanics, Celestial Mechanics & Dynamical Astronomy, 52 (1991), 221-240.  doi: 10.1007/BF00048485.  Google Scholar [5] E. Hairer, Variable time step integration with symplectic methods, Applied Numerical Mathematics, 25 (1997), 219-227.  doi: 10.1016/S0168-9274(97)00061-5.  Google Scholar [6] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition, Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2006.  Google Scholar [7] E. Hairer and C. Lubich, Long-term analysis of a variational integrator for charged-particle dynamics in a strong magnetic field, preprint, Available from: https://na.uni-tuebingen.de/pub/lubich/papers/adiabatic.pdf. Google Scholar [8] Y. He, Y. J. Sun, J. Liu and H. Qin, Volume-preserving algorithms for charged particle dynamics, Journal of Computational Physics, 281 (2015), 135-147.  doi: 10.1016/j.jcp.2014.10.032.  Google Scholar [9] Y. He, Y. J. Sun, R. L. Zhang, Y. L. Wang, J. Liu and H. Qin, High order volume-preserving algorithms for relativistic charged particles in general electromagnetic fields, Physics of Plasmas, 23 (2016), 092109. doi: 10.1063/1.4962677.  Google Scholar [10] Y. He, Z. Q. Zhou, Y. J. Sun, J. Liu and H. Qin, Explicit $K$-symplectic algorithms for charged particle dynamics, Physics Letters A, 381 (2017), 568-573.  doi: 10.1016/j.physleta.2016.12.031.  Google Scholar [11] W. Z. Huang and B. Leimkuhler, The adaptive verlet method, SIAM Journal on Scientific Computing, 18 (1997), 239-256.  doi: 10.1137/S1064827595284658.  Google Scholar [12] J. D. Jackson, From Lorenz to Coulomb and other explicit gauge transformations, American Journal of Physics, 70 (2002), 917-928.  doi: 10.1119/1.1491265.  Google Scholar [13] C. Knapp, A. Kendl, A. Koskela and A. Ostermann, Splitting methods for time integration of trajectories in combined electric and magnetic fields, Physical Review E, 92 (2015), 063310, 13 pp.  Google Scholar [14] M. Kretzschmar, Single particle motion in a Penning trap: Description in the classical canonical formalism, Physica Scripta, 46 (1992), 544-554.  doi: 10.1088/0031-8949/46/6/011.  Google Scholar [15] P. Langevin, Sur la théorie du mouvement brownien, C. R. Acad. Sci. (Paris), 146 (1908), 530-533.   Google Scholar [16] J. Liu, H. Qin, Y. Wang and et al., Largest particle simulations downgrade the runaway electron risk for Iter, arXiv: 1611.02362. Google Scholar [17] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2682-6.  Google Scholar [18] H. Qin and R. C. Davidson, An exact magnetic-moment invariant of charged-particle gyromotion, Physical Review Letters, 96 (2006), 085003. doi: 10.1103/PhysRevLett.96.085003.  Google Scholar [19] H. Qin and X. Y. Guan, Variational symplectic integrator for long-time simulations of the guiding-center motion of charged particles in general magnetic fields, Physical Review Letters, 100 (2008), 035006. doi: 10.1103/PhysRevLett.100.035006.  Google Scholar [20] H. Qin, X. Y. Guan and W. M. Tang, Variational symplectic algorithm for guiding center dynamics and its application in tokamak geometry, Physics of Plasmas, 16 (2009), 042510. doi: 10.1063/1.3099055.  Google Scholar [21] S. Reich, Backward error analysis for numerical integrators, SIAM Journal on Numerical Analysis, 36 (1999), 1549-1570.  doi: 10.1137/S0036142997329797.  Google Scholar [22] A. S. Richardson and J. M. Finn, Symplectic integrators with adaptive time steps, Plasma Physics and Controlled Fusion, 54 (2012), 96-100.  doi: 10.1088/0741-3335/54/1/014004.  Google Scholar [23] C. C. Rodegheri, K. Blaum, H. Kracke, S. Kreim, A. Mooser, W. Quint, S. Ulmer and J. Walz, An experiment for the direct determination of the g-factor of a single proton in a Penning trap, New Journal of Physics, 14 (2012), 063011. doi: 10.1088/1367-2630/14/6/063011.  Google Scholar [24] J. Schmitt and M. Leok, Adaptive variational integrators, arXiv: 1709.01975. Google Scholar [25] Y. Y. Shi, Y. J. Sun, Y. He, H. Qin and J. Liu, Symplectic integrators with adaptive time step applied to runaway electron dynamics, Numerical Algorithms, 81 (2019), 1295-1309.  doi: 10.1007/s11075-018-0636-6.  Google Scholar [26] M. Toggweiler, A. Adelmann, P. Arbenz and J. J. Yang, A novel adaptive time stepping variant of the Boris–Buneman integrator for the simulation of particle accelerators with space charge, Journal of Computational Physics, 273 (2014), 255-267.  doi: 10.1016/j.jcp.2014.05.008.  Google Scholar [27] Y. L. Wang, J. Liu and H. Qin, Lorentz covariant canonical symplectic algorithms for dynamics of charged particles, Physics of Plasmas, 23 (2016), 122513. doi: 10.1063/1.4972824.  Google Scholar [28] R. L. Zhang, Y. L. Wang, Y. He, J. Y. Xiao, J. Liu, H. Qin and Y. F. Tang, Explicit symplectic algorithms based on generating functions for relativistic charged particle dynamics in time-dependent electromagnetic field, Physics of Plasmas, 25 (2018), 022117. doi: 10.1063/1.5012767.  Google Scholar [29] Z. Q. Zhou, Y. He, Y. J. Sun, J. Liu and H. Qin, Explicit symplectic methods for solving charged particle trajectories, Physics of Plasmas, 24 (2017), 052507. doi: 10.1063/1.4982743.  Google Scholar

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##### References:
 [1] G. Benettin and P. Sempio, Adiabatic invariants and trapping of a point charge in a strong nonuniform magnetic field, Nonlinearity, 7 (1994), 281-303.  doi: 10.1088/0951-7715/7/1/014.  Google Scholar [2] M. P. Calvo and J. M. Sanz-Serna, The Development of variable-step symplectic integrators, with application to the two-body problem, SIAM Journal on Scientific Computing, 14 (1993), 936-952.  doi: 10.1137/0914057.  Google Scholar [3] H. Dreicer, Electron and ion runaway in a fully ionized gas, Physical Review, 115 (1959), 238-249.  doi: 10.1103/PhysRev.115.238.  Google Scholar [4] B. Gladman, M. Duncan and J. Candy, Symplectic integrators for long-term integrations in celestial mechanics, Celestial Mechanics & Dynamical Astronomy, 52 (1991), 221-240.  doi: 10.1007/BF00048485.  Google Scholar [5] E. Hairer, Variable time step integration with symplectic methods, Applied Numerical Mathematics, 25 (1997), 219-227.  doi: 10.1016/S0168-9274(97)00061-5.  Google Scholar [6] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition, Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2006.  Google Scholar [7] E. Hairer and C. Lubich, Long-term analysis of a variational integrator for charged-particle dynamics in a strong magnetic field, preprint, Available from: https://na.uni-tuebingen.de/pub/lubich/papers/adiabatic.pdf. Google Scholar [8] Y. He, Y. J. Sun, J. Liu and H. Qin, Volume-preserving algorithms for charged particle dynamics, Journal of Computational Physics, 281 (2015), 135-147.  doi: 10.1016/j.jcp.2014.10.032.  Google Scholar [9] Y. He, Y. J. Sun, R. L. Zhang, Y. L. Wang, J. Liu and H. Qin, High order volume-preserving algorithms for relativistic charged particles in general electromagnetic fields, Physics of Plasmas, 23 (2016), 092109. doi: 10.1063/1.4962677.  Google Scholar [10] Y. He, Z. Q. Zhou, Y. J. Sun, J. Liu and H. Qin, Explicit $K$-symplectic algorithms for charged particle dynamics, Physics Letters A, 381 (2017), 568-573.  doi: 10.1016/j.physleta.2016.12.031.  Google Scholar [11] W. Z. Huang and B. Leimkuhler, The adaptive verlet method, SIAM Journal on Scientific Computing, 18 (1997), 239-256.  doi: 10.1137/S1064827595284658.  Google Scholar [12] J. D. Jackson, From Lorenz to Coulomb and other explicit gauge transformations, American Journal of Physics, 70 (2002), 917-928.  doi: 10.1119/1.1491265.  Google Scholar [13] C. Knapp, A. Kendl, A. Koskela and A. Ostermann, Splitting methods for time integration of trajectories in combined electric and magnetic fields, Physical Review E, 92 (2015), 063310, 13 pp.  Google Scholar [14] M. Kretzschmar, Single particle motion in a Penning trap: Description in the classical canonical formalism, Physica Scripta, 46 (1992), 544-554.  doi: 10.1088/0031-8949/46/6/011.  Google Scholar [15] P. Langevin, Sur la théorie du mouvement brownien, C. R. Acad. Sci. (Paris), 146 (1908), 530-533.   Google Scholar [16] J. Liu, H. Qin, Y. Wang and et al., Largest particle simulations downgrade the runaway electron risk for Iter, arXiv: 1611.02362. Google Scholar [17] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2682-6.  Google Scholar [18] H. Qin and R. C. Davidson, An exact magnetic-moment invariant of charged-particle gyromotion, Physical Review Letters, 96 (2006), 085003. doi: 10.1103/PhysRevLett.96.085003.  Google Scholar [19] H. Qin and X. Y. Guan, Variational symplectic integrator for long-time simulations of the guiding-center motion of charged particles in general magnetic fields, Physical Review Letters, 100 (2008), 035006. doi: 10.1103/PhysRevLett.100.035006.  Google Scholar [20] H. Qin, X. Y. Guan and W. M. Tang, Variational symplectic algorithm for guiding center dynamics and its application in tokamak geometry, Physics of Plasmas, 16 (2009), 042510. doi: 10.1063/1.3099055.  Google Scholar [21] S. Reich, Backward error analysis for numerical integrators, SIAM Journal on Numerical Analysis, 36 (1999), 1549-1570.  doi: 10.1137/S0036142997329797.  Google Scholar [22] A. S. Richardson and J. M. Finn, Symplectic integrators with adaptive time steps, Plasma Physics and Controlled Fusion, 54 (2012), 96-100.  doi: 10.1088/0741-3335/54/1/014004.  Google Scholar [23] C. C. Rodegheri, K. Blaum, H. Kracke, S. Kreim, A. Mooser, W. Quint, S. Ulmer and J. Walz, An experiment for the direct determination of the g-factor of a single proton in a Penning trap, New Journal of Physics, 14 (2012), 063011. doi: 10.1088/1367-2630/14/6/063011.  Google Scholar [24] J. Schmitt and M. Leok, Adaptive variational integrators, arXiv: 1709.01975. Google Scholar [25] Y. Y. Shi, Y. J. Sun, Y. He, H. Qin and J. Liu, Symplectic integrators with adaptive time step applied to runaway electron dynamics, Numerical Algorithms, 81 (2019), 1295-1309.  doi: 10.1007/s11075-018-0636-6.  Google Scholar [26] M. Toggweiler, A. Adelmann, P. Arbenz and J. J. Yang, A novel adaptive time stepping variant of the Boris–Buneman integrator for the simulation of particle accelerators with space charge, Journal of Computational Physics, 273 (2014), 255-267.  doi: 10.1016/j.jcp.2014.05.008.  Google Scholar [27] Y. L. Wang, J. Liu and H. Qin, Lorentz covariant canonical symplectic algorithms for dynamics of charged particles, Physics of Plasmas, 23 (2016), 122513. doi: 10.1063/1.4972824.  Google Scholar [28] R. L. Zhang, Y. L. Wang, Y. He, J. Y. Xiao, J. Liu, H. Qin and Y. F. Tang, Explicit symplectic algorithms based on generating functions for relativistic charged particle dynamics in time-dependent electromagnetic field, Physics of Plasmas, 25 (2018), 022117. doi: 10.1063/1.5012767.  Google Scholar [29] Z. Q. Zhou, Y. He, Y. J. Sun, J. Liu and H. Qin, Explicit symplectic methods for solving charged particle trajectories, Physics of Plasmas, 24 (2017), 052507. doi: 10.1063/1.4982743.  Google Scholar
Illustration of adaptive time step strategy
Illustration of backward analysis for adaptive symplectic method
Particle trajectory under uniform magnetic field. (a) Trajectory computed by implicit midpoint rule with fixed time step. (b) Trajectory computed by implicit midpoint rule with adaptive time step
Particle trajectory under magnetic bottle. (a) Trajectory computed by implicit midpoint rule with fixed time step. (b) Trajectory computed by implicit midpoint rule with adaptive time step
Relative error of Hamiltonian by adaptive implicit midpoint rule with $\Delta t = \Delta \tau/B = 0.05 /B$
Absolute error of magnetic moment $\mu$ for adaptive implicit midpoint rule with $\Delta t = \Delta \tau /B = 0.05/B$
Comparison of implicit midpoint rule with fixed and adaptive time steps in simulating a single charged particle motion in magnetic bottle up to $T = 100$
Particle trajectory under asymmetric magnetic field. (a) Trajectory computed by implicit midpoint rule with fixed step size. (b) Trajectory computed by variable step size
Relative error of Hamiltonian for fixed midpoint rule with $\Delta t = 0.012$ and adaptive midpoint rule with $\Delta t = \Delta \tau /B = 0.012/B$
Absolute error of magnetic moment $\mu$ for fixed implicit midpoint rule with $\Delta t = 0.012$ and adaptive midpoint rule with $\Delta t = \Delta \tau /B = 0.012/B$
Comparison of the implicit midpoint rule with fixed and adaptive time steps to simulate the charged particle motion in asymmetric magnetic field up to $T = 100$
Change of time step $\Delta t$ with $\Delta\tau = 1.98$
Relative error of Hamiltonian for adaptive time steps method with $\Delta t = \Delta \tau/B = 1.98/B$
Absolute error of magnetic moment $\mu$ for adaptive time steps method with $\Delta t = \Delta \tau/B = 1.98/B$
The first and the last gyro-orbits of a single particle motion to $T = 1000$. (a) and (b): fixed time step Gauss method. (c) and (d): adaptive time step Gauss method
Relative error of Hamiltonian for adaptive implicit midpoint rule and Gauss method with $\Delta t = \Delta \tau\gamma/B = 0.06\gamma/B$. Here, $B_0 = 1, E_0 = 0.005$)
Relative error of Hamiltonian for adaptive time steps method with $\Delta t = \Delta \tau\gamma/B = 0.8\gamma/B$ ($E_0 = -2$, $B_0 = -5.3$)
The poloidal cross-section of distribution evolution of $10^5$ runaway particles. (a) $T = 0$, (b) $T = 2\cdot 10^{10}$, (c) $T = 5\cdot 10^{10}$
The poloidal cross-section of distribution evolution of $10^5$ runaway particles. (a) $T = 0$, (b) $T = 2\cdot 10^{10}$, (c) $T = 5\cdot 10^{10}$
Comparison of fixed and adaptive time step methods to simulate the same acceleration process for an electron up to $T = 2\cdot 10^6$ with $B_0 = 5.3\; \text{T}$, $E_0 = 200\; \text{V/m}$, and the initial values are $\mathbf{x}_0 = [23296.7,\; 0,\; 0]^\top$, $\mathbf{P}_0 = [3.3875,\; 165.0318,\; -1831.295]^\top$
Dimensionless way of physical quantities. Here, $m_0$ is the rest mass of a particle, $e$ is the elementary charge, $c$ is the speed of light, and $B_0$ is the given reference magnetic field. (In this paper $m_0 = 9.1\times10^{-31}$, $e = 1.6\times10^{-19}$, $c = 3.0\times10^{8}$.)
 Quantities Symbols Non-relativistic Relativistic Units Units Time $t$ $m_0/(eB_0)$ $m_0/(eB_0)$ Position $\mathbf{x}$ $m_0/(eB_0)$ $m_0c/(eB_0)$ Velocity $\mathbf{v}$ $1$ $c$ Momentum $\mathbf{p}$ $m_0$ $m_0c$ Canonical Momentum $\mathbf{P}$ $m_0$ $m_0c$ Electric field $\mathbf{E}$ $B_0$ $B_0c$ Magnetic field $\mathbf{B}$ $B_0$ $B_0$ Vector field $\mathbf{A}$ $m_0/e$ $m_0c/e$ Scalar field $\varphi$ $m_0/e$ $m_0c^2/e$ Hamiltonian $H$ $m_0$ $m_0c^2$
 Quantities Symbols Non-relativistic Relativistic Units Units Time $t$ $m_0/(eB_0)$ $m_0/(eB_0)$ Position $\mathbf{x}$ $m_0/(eB_0)$ $m_0c/(eB_0)$ Velocity $\mathbf{v}$ $1$ $c$ Momentum $\mathbf{p}$ $m_0$ $m_0c$ Canonical Momentum $\mathbf{P}$ $m_0$ $m_0c$ Electric field $\mathbf{E}$ $B_0$ $B_0c$ Magnetic field $\mathbf{B}$ $B_0$ $B_0$ Vector field $\mathbf{A}$ $m_0/e$ $m_0c/e$ Scalar field $\varphi$ $m_0/e$ $m_0c^2/e$ Hamiltonian $H$ $m_0$ $m_0c^2$
The coefficients of the 4th order Gauss method
 $\frac{6-2\sqrt{3}}{12}$ $\frac{1}{4}$ $\frac{3-2\sqrt{3}}{12}$ $\frac{6+2\sqrt{3}}{12}$ $\frac{3+2\sqrt{3}}{12}$ $\frac{1}{4}$ $\frac{1}{2}$ $\frac{1}{2}$
 $\frac{6-2\sqrt{3}}{12}$ $\frac{1}{4}$ $\frac{3-2\sqrt{3}}{12}$ $\frac{6+2\sqrt{3}}{12}$ $\frac{3+2\sqrt{3}}{12}$ $\frac{1}{4}$ $\frac{1}{2}$ $\frac{1}{2}$
Time iterations steps for motion over various gyro-period by fixed and adaptive time step Gauss methods
 period 1st 70th 120th 150th fixed 100 451 2134 5470 adaptive 100 100 100 100
 period 1st 70th 120th 150th fixed 100 451 2134 5470 adaptive 100 100 100 100
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