American Institute of Mathematical Sciences

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

Study of adaptive symplectic methods for simulating charged particle dynamics

Yanyan Shi 1,2, , Yajuan Sun 1,2,, , Yulei Wang 3,4, and  Jian Liu 3,4,

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.

Keywords: Charged particle dynamics, symplectic methods, adaptive time step, Penning trap, tokamaks.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
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

show all references

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

Figure 1.  Illustration of adaptive time step strategy
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Illustration of backward analysis for adaptive symplectic method
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Relative error of Hamiltonian by adaptive implicit midpoint rule with $ \Delta t = \Delta \tau/B = 0.05 /B $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Absolute error of magnetic moment $ \mu $ for adaptive implicit midpoint rule with $ \Delta t = \Delta \tau /B = 0.05/B $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Particle trajectory under asymmetric magnetic field. (a) Trajectory computed by implicit midpoint rule with fixed step size. (b) Trajectory computed by variable step size
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12.  Change of time step $ \Delta t $ with $ \Delta\tau = 1.98 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 13.  Relative error of Hamiltonian for adaptive time steps method with $ \Delta t = \Delta \tau/B = 1.98/B $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 14.  Absolute error of magnetic moment $ \mu $ for adaptive time steps method with $ \Delta t = \Delta \tau/B = 1.98/B $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 15.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 16.  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 $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 17.  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 $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 18.  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} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 19.  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} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 20.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  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 $
Table Options

Download as excel

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 $
Table 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} $
Table Options

Download as excel

$ \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} $
Table 3.  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
Table Options

Download as excel

period 1st 70th 120th 150th
fixed 100 451 2134 5470
adaptive 100 100 100 100
[1]

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
[2]

Raffaele D'Ambrosio, Martina Moccaldi, Beatrice Paternoster. Numerical preservation of long-term dynamics by stochastic two-step methods. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2763-2773. doi: 10.3934/dcdsb.2018105
[3]

Mihai Bostan. On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 339-371. doi: 10.3934/dcdsb.2015.20.339
[4]

Mathieu Lutz. Application of Lie transform techniques for simulation of a charged particle beam. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 185-221. doi: 10.3934/dcdss.2015.8.185
[5]

Lijin Wang, Jialin Hong. Generating functions for stochastic symplectic methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1211-1228. doi: 10.3934/dcds.2014.34.1211
[6]

Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105
[7]

Andrew J. Steyer, Erik S. Van Vleck. Underlying one-step methods and nonautonomous stability of general linear methods. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2859-2877. doi: 10.3934/dcdsb.2018108
[8]

Alexandre Mouton. Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. Kinetic & Related Models, 2009, 2 (2) : 251-274. doi: 10.3934/krm.2009.2.251
[9]

Giacomo Frassoldati, Luca Zanni, Gaetano Zanghirati. New adaptive stepsize selections in gradient methods. Journal of Industrial & Management Optimization, 2008, 4 (2) : 299-312. doi: 10.3934/jimo.2008.4.299
[10]

Cesare Bracco, Annalisa Buffa, Carlotta Giannelli, Rafael Vázquez. Adaptive isogeometric methods with hierarchical splines: An overview. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 241-261. doi: 10.3934/dcds.2019010
[11]

Qifeng Cheng, Xue Han, Tingting Zhao, V S Sarma Yadavalli. Improved particle swarm optimization and neighborhood field optimization by introducing the re-sampling step of particle filter. Journal of Industrial & Management Optimization, 2019, 15 (1) : 177-198. doi: 10.3934/jimo.2018038
[12]

Tomáš Roubíček, Giuseppe Tomassetti. Dynamics of charged elastic bodies under diffusion at large strains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019234
[13]

Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1811-1827. doi: 10.3934/dcds.2014.34.1811
[14]

Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077
[15]

Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185
[16]

Pablo G. Barrientos, Artem Raibekas. Robustly non-hyperbolic transitive symplectic dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 5993-6013. doi: 10.3934/dcds.2018259
[17]

Yoonsang Lee, Bjorn Engquist. Variable step size multiscale methods for stiff and highly oscillatory dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1079-1097. doi: 10.3934/dcds.2014.34.1079
[18]

Tingting Wu, Yufei Yang, Huichao Jing. Two-step methods for image zooming using duality strategies. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 209-225. doi: 10.3934/naco.2014.4.209
[19]

Angelamaria Cardone, Dajana Conte, Beatrice Paternoster. Two-step collocation methods for fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2709-2725. doi: 10.3934/dcdsb.2018088
[20]

Miguel A. Dumett, Roberto Cominetti. On the stability of an adaptive learning dynamics in traffic games. Journal of Dynamics & Games, 2018, 5 (4) : 265-282. doi: 10.3934/jdg.2018017

 Impact Factor: 

Tools

Metrics

  • PDF downloads (36)
  • HTML views (37)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences