{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:10:26Z","timestamp":1760148626421,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T00:00:00Z","timestamp":1684972800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Research Funds for the Central Universities","award":["FRF-TP-20-068A1"],"award-info":[{"award-number":["FRF-TP-20-068A1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We propose explicit K-symplectic and explicit symplectic-like methods for the charged particle system in a general strong magnetic field. The K-symplectic methods are also symmetric. The charged particle system can be expressed both in a canonical and a non-canonical Hamiltonian system. If the three components of the magnetic field can be integrated in closed forms, we construct explicit K-symplectic methods for the non-canonical charged particle system; otherwise, explicit symplectic-like methods can be constructed for the canonical charged particle system. The symplectic-like methods are constructed by extending the original phase space and obtaining the augmented separable Hamiltonian, and then by using the splitting method and the midpoint permutation. The numerical experiments have shown that compared with the higher order implicit Runge-Kutta method, the explicit K-symplectic and explicit symplectic-like methods have obvious advantages in long-term energy conservation and higher computational efficiency. It is also shown that the influence of the parameter \u03b5 in the general strong magnetic field on the Runge-Kutta method is bigger than the two kinds of symplectic methods.<\/jats:p>","DOI":"10.3390\/sym15061146","type":"journal-article","created":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T01:36:28Z","timestamp":1684978588000},"page":"1146","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Explicit K-Symplectic and Symplectic-like Methods for Charged Particle System in General Magnetic Field"],"prefix":"10.3390","volume":"15","author":[{"given":"Yulan","family":"Lu","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junbin","family":"Yuan","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haoyang","family":"Tian","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhengwei","family":"Qin","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Siyuan","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongji","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"092109","DOI":"10.1063\/1.4962677","article-title":"High order volume-preserving algorithms for relativistic charged particles in general electromagnetic fields","volume":"23","author":"He","year":"2016","journal-title":"Phys. Plasmas"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"013205","DOI":"10.1103\/PhysRevE.94.013205","article-title":"Explicit symplectic algorithms based on generating functions for charged particle dynamics","volume":"94","author":"Zhang","year":"2016","journal-title":"Phys. Rev. E."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"022117","DOI":"10.1063\/1.5012767","article-title":"Explicit symplectic algorithms based on generating functions for relativistic charged particle dynamics in time-dependent electromagnetic field","volume":"25","author":"Zhang","year":"2018","journal-title":"Phys. Plasmas"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Tang, R., and Li, D. (2021). On Symmetric Methods for Charged Particle Dynamics. Symmetry, 13.","DOI":"10.3390\/sym13091626"},{"key":"ref_5","unstructured":"Feng, K. (1995). Collected Works of Feng Kang (II), National Defence Industry Press."},{"key":"ref_6","unstructured":"Feng, K. (1985). Proceedings of 1984 Beijing Symposium on Differential Geometry and Differential Equations, Science Press."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Feng, K., and Qin, M.Z. (2009). Symplectic Geometric Algorithms for Hamiltonian System, Springer.","DOI":"10.1007\/978-3-642-01777-3"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Sanz-Serna, J.M., and Calvo, M.P. (1994). Numerical Hamiltonian Problems, Chapman and Hall.","DOI":"10.1007\/978-1-4899-3093-4"},{"key":"ref_9","first-page":"17","article-title":"Symplectic Methods for the Ablowitz-Ladik Model","volume":"82","author":"Tang","year":"1997","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Brugnano, L., and Iavernaro, F. (2016). Line Integral Methods for Conservative Problems, Chapman and Hall\/CRC.","DOI":"10.1201\/b19319"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.cnsns.2017.12.018","article-title":"A class of energy-conserving Hamiltonian boundary value methods for nonlinear Schr\u00f6dinger equation with wave operator","volume":"60","author":"Brugnano","year":"2018","journal-title":"Commun. Nonlin. Sci. Numer. Simulat."},{"key":"ref_12","first-page":"265","article-title":"Construction of volume-preserving difference schemes for source-free systems via generating functions","volume":"12","author":"Shang","year":"1994","journal-title":"J. Comput. Math."},{"key":"ref_13","first-page":"1172","article-title":"Generating functions for volume-preserving mappings and Hamilton-Jacobi equations for source-free dynamical systems","volume":"37","author":"Shang","year":"1994","journal-title":"Sci. China Ser. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1088\/0951-7715\/3\/2\/001","article-title":"Symplectic Integration of Hamiltonian Systems","volume":"3","author":"Channell","year":"1990","journal-title":"Nonlinearity"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Hairer, E., Lubich, C., and Wanner, G. (2002). Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer.","DOI":"10.1007\/978-3-662-05018-7"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kosinski, P., and Maslanka, P. (2020). Relativistic symmetries and Hamiltonian formalism. Symmetry, 12.","DOI":"10.3390\/sym12111810"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zajac, M., Sardon, C., and Ragnisco, O. (2023). Time-dependent Hamiltonian mechanics on a locally conformal symplectic manifold. Symmetry, 15.","DOI":"10.3390\/sym15040843"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Arnold, V.I. (1978). Mathematical Methods of Classical Mechanics, Springer.","DOI":"10.1007\/978-1-4757-1693-1"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Albosaily, S., Mohammed, W.W., Aiyashi, M.A., and Abdelrahman, M.A.E. (2020). Exact solutions of the (2 + 1)-dimensional stochastic chiral nonlinear Schr\u00f6dinger equation. Symmetry, 12.","DOI":"10.3390\/sym12111874"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Shah, N.A., Agarwal, P., Chung, J.D., El-Zahar, E.R., and Hamed, Y.S. (2020). Analysis of optical solitons for nonlinear Schr\u00f6dinger equation with detuning term by iterative transform method. Symmetry, 12.","DOI":"10.3390\/sym12111850"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1006\/jcph.2001.6733","article-title":"Splitting Methods for Non-autonomous Hamiltonian Equations","volume":"170","author":"Blanes","year":"2001","journal-title":"J. Comput. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1016\/j.physleta.2016.12.031","article-title":"Explicit K-symplectic algorithms for charged particle dynamics","volume":"381","author":"He","year":"2017","journal-title":"Phys. Lett. A"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zhu, B.B., and Zhao, Y.L. (2021). Symplectic all-at-once method for Hamiltonian systems. Symmetry, 13.","DOI":"10.3390\/sym13101930"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1485","DOI":"10.1007\/s11075-019-00708-8","article-title":"Symplectic simulation of dark solitons motion for nonlinear Schr\u00f6dinger equation","volume":"81","author":"Zhu","year":"2019","journal-title":"Numer. Algorithms"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/s10569-014-9597-9","article-title":"Explicit methods in extended phase space for inseparable Hamiltonian problems","volume":"121","author":"Pihajoki","year":"2015","journal-title":"Celest. Mech. Dyn. Astr."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"043303","DOI":"10.1103\/PhysRevE.94.043303","article-title":"Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance","volume":"94","author":"Tao","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1850006","DOI":"10.1142\/S0129183118500067","article-title":"An optimized Forest-Ruth-like algorithm in extended phase space","volume":"29","author":"Wu","year":"2018","journal-title":"Int. J. Mod. Phys. C"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1090\/mcom\/3778","article-title":"Semiexplicit symplectic integrators for non-separable Hamiltonian systems","volume":"92","author":"Jayawardana","year":"2023","journal-title":"Math. Comput."},{"key":"ref_29","unstructured":"Ohsawa, T. (2022). Preservation of Quadratic Invariants by Semiexplicit Symplectic Integrators for Non-separable Hamiltonian Systems. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1137\/0705041","article-title":"On the construction and comparison of difference schemes","volume":"5","author":"Strang","year":"1968","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/S0377-0427(01)00492-7","article-title":"Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nystrom methods","volume":"142","author":"Blanes","year":"2002","journal-title":"J. Comput. Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1090\/S0025-5718-1964-0159424-9","article-title":"Implicit Runge-Kutta Processes","volume":"18","author":"Butcher","year":"1964","journal-title":"Math. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Hairer, E., N\u00f8rsett, S.P., and Wanner, G. (1987). Solving Ordinary Differential Equation I: Nonstiff Problems, Springer.","DOI":"10.1007\/978-3-662-12607-3"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1146\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:41:31Z","timestamp":1760125291000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1146"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,25]]},"references-count":33,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["sym15061146"],"URL":"https:\/\/doi.org\/10.3390\/sym15061146","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,5,25]]}}}