Skip to main content
SpringerLink
Log in

A two-step symmetric method for charged-particle dynamics in a normal or strong magnetic field

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

The study of the long time conservation for numerical methods poses interesting and challenging questions from the point of view of geometric integration. In this paper, we analyze the long time energy and magnetic moment conservations of two-step symmetric methods for charged-particle dynamics. A two-step symmetric method is proposed and its long time behaviour is shown not only in a normal magnetic field but also in a strong magnetic field. The approaches to dealing with these two cases are based on the backward error analysis and modulated Fourier expansion, respectively. It is obtained from the analysis that the method has better long time conservations than the variational method which was researched recently in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Boris, J.P.: Relativistic plasma simulation-optimization of a hybrid code. In: Proceeding of Fourth Conference on Numerical Simulations of Plasmas, pp. 3–67 (1970)

  2. Cohen, D., Gauckler, L.: One-stage exponential integrators for nonlinear Schrödinger equations over long times. BIT 52, 877–903 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cohen, D., Hairer, E., Lubich, C.: Numerical energy conservation for multi-frequency oscillatory differential equations. BIT 45, 287–305 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cohen, D., Hairer, E., Lubich, C.: Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations. Numer. Math. 110, 113–143 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gauckler, L., Hairer, E., Lubich, C.: Long-term analysis of semilinear wave equations with slowly varying wave speed. Commun. Part. Differ. Equ. 41, 1934–1959 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hairer, E., Lubich, C.: Long-term analysis of the Störmer–Verlet method for Hamiltonian systems with a solution-dependent high frequency. Numer. Math. 134, 119–138 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hairer, E., Lubich, C.: Energy behaviour of the Boris method for charged-particle dynamics. BIT 58, 969–979 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hairer, E., Lubich, C.: Symmetric multistep methods for charged-particle dynamics. SMAI J. Comput. Math. 3, 205–218 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hairer, E., Lubich, C.: Long-term analysis of a variational integrator for charged-particle dynamics in a strong magnetic field. Numer. Math. 144, 787–809 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  12. He, Y., Sun, Y., Liu, J., Qin, H.: Volume-preserving algorithms for charged particle dynamics. J. Comput. Phys. 281, 135–147 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. He, Y., Zhou, Z., Sun, Y., Liu, J., Qin, H.: Explicit K-symplectic algorithms for charged particle dynamics. Phys. Lett. A 381, 568–573 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, T., Wang, B.: Efficient energy-preserving methods for charged-particle dynamics. Appl. Math. Comput. 361, 703–714 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, T., Wang, B.: Arbitrary-order energy-preserving methods for charged-particle dynamics. Appl. Math. Lett. 100, 106050 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Qin, H., Zhang, S., Xiao, J., Liu, J., Sun, Y., Tang, W.M.: Why is Boris algorithm so good? Phys. Plasmas 20, 084503 (2013)

    Article  Google Scholar 

  17. Tao, M.: Explicit high-order symplectic integrators for charged particles in general electromagnetic fields. J. Comput. Phys. 327, 245–251 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, B., Wu, X.: Long-time momentum and actions behaviour of energy-preserving methods for semilinear wave equations via spatial spectral semi-discretizations. Adv. Comput. Math.  45, 2921–2952 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Webb, S.D.: Symplectic integration of magnetic systems. J. Comput. Phys. 270, 570–576 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, R., Qin, H., Tang, Y., Liu, J., He, Y., Xiao, J.: Explicit symplectic algorithms based on generating functions for charged particle dynamics. Phys. Rev. E 94, 013205 (2016)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are sincerely thankful to the anonymous reviewers for their valuable suggestions, which help improve the presentation of the manuscript significantly. The first author is grateful to Professor Christian Lubich for the discussions and helpful comments on the long term analysis of methods for charged-particle dynamics.

Funding

The work is supported by the Alexander von Humboldt Foundation, by the Natural Science Foundation of Shandong Province (Outstanding Youth Foundation) under Grant ZR2017JL003, by the National Natural Science Foundation of China under Grants 11571302, 11671200, by the Foundation of Innovative Science and Technology for youth in universities of Shandong Province under Grant 2019KJI001, and the project of Qingtan Scholars of Zaozhuang University.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shannxi, People’s Republic of China

    Bin Wang

  2. Mathematisches Institut, University of Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    Bin Wang

  3. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Xinyuan Wu

  4. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, People’s Republic of China

    Xinyuan Wu

  5. School of Mathematics and Statistics, Zaozhuang University, Zaozhuang, 277160, People’s Republic of China

    Yonglei Fang

Authors
  1. Bin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinyuan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yonglei Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bin Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, B., Wu, X. & Fang, Y. A two-step symmetric method for charged-particle dynamics in a normal or strong magnetic field. Calcolo 57, 29 (2020). https://doi.org/10.1007/s10092-020-00377-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10092-020-00377-3

Keywords

Mathematics Subject Classification

Search

Navigation