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.
Similar content being viewed by others
References
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)
Cohen, D., Gauckler, L.: One-stage exponential integrators for nonlinear Schrödinger equations over long times. BIT 52, 877–903 (2012)
Cohen, D., Hairer, E., Lubich, C.: Numerical energy conservation for multi-frequency oscillatory differential equations. BIT 45, 287–305 (2005)
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)
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)
Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)
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)
Hairer, E., Lubich, C.: Energy behaviour of the Boris method for charged-particle dynamics. BIT 58, 969–979 (2018)
Hairer, E., Lubich, C.: Symmetric multistep methods for charged-particle dynamics. SMAI J. Comput. Math. 3, 205–218 (2017)
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)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)
He, Y., Sun, Y., Liu, J., Qin, H.: Volume-preserving algorithms for charged particle dynamics. J. Comput. Phys. 281, 135–147 (2015)
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)
Li, T., Wang, B.: Efficient energy-preserving methods for charged-particle dynamics. Appl. Math. Comput. 361, 703–714 (2019)
Li, T., Wang, B.: Arbitrary-order energy-preserving methods for charged-particle dynamics. Appl. Math. Lett. 100, 106050 (2020)
Qin, H., Zhang, S., Xiao, J., Liu, J., Sun, Y., Tang, W.M.: Why is Boris algorithm so good? Phys. Plasmas 20, 084503 (2013)
Tao, M.: Explicit high-order symplectic integrators for charged particles in general electromagnetic fields. J. Comput. Phys. 327, 245–251 (2016)
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)
Webb, S.D.: Symplectic integration of magnetic systems. J. Comput. Phys. 270, 570–576 (2014)
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)
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
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
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
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10092-020-00377-3
Keywords
- Charged-particle dynamics
- Two-step symmetric methods
- Backward error analysis
- Modulated Fourier expansion