{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:26:25Z","timestamp":1760235985688,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,14]],"date-time":"2021-10-14T00:00:00Z","timestamp":1634169600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11901564"],"award-info":[{"award-number":["11901564"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The all-at-once technique has attracted many researchers\u2019 interest in recent years. In this paper, we combine this technique with a classical symplectic and symmetric method for solving Hamiltonian systems. The solutions at all time steps are obtained at one-shot. In order to reduce the computational cost of solving the all-at-once system, a fast algorithm is designed. Numerical experiments of Hamiltonian systems with degrees of freedom n\u22643 are provided to show that our method is more efficient than the classical symplectic method.<\/jats:p>","DOI":"10.3390\/sym13101930","type":"journal-article","created":{"date-parts":[[2021,10,14]],"date-time":"2021-10-14T23:02:16Z","timestamp":1634252536000},"page":"1930","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Symplectic All-at-Once Method for Hamiltonian Systems"],"prefix":"10.3390","volume":"13","author":[{"given":"Bei-Bei","family":"Zhu","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yong-Liang","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hairer, E., Lubich, C., and Wanner, G. 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