{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:32:17Z","timestamp":1760239937077,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,1,28]],"date-time":"2019-01-28T00:00:00Z","timestamp":1548633600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A family of explicit symplectic partitioned Runge-Kutta methods are derived with effective order 3 for the numerical integration of separable Hamiltonian systems. The proposed explicit methods are more efficient than existing symplectic implicit Runge-Kutta methods. A selection of numerical experiments on separable Hamiltonian system confirming the efficiency of the approach is also provided with good energy conservation.<\/jats:p>","DOI":"10.3390\/sym11020142","type":"journal-article","created":{"date-parts":[[2019,1,29]],"date-time":"2019-01-29T03:40:55Z","timestamp":1548733255000},"page":"142","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Symplectic Effective Order Numerical Methods for Separable Hamiltonian Systems"],"prefix":"10.3390","volume":"11","author":[{"given":"Junaid","family":"Ahmad","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Engineering and Technology, Lahore 54890, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1709-1159","authenticated-orcid":false,"given":"Yousaf","family":"Habib","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 45550, Pakistan"},{"name":"School of Natural Sciences, Department of Mathematics, National University of Sciences and Technology, Islamabad 44000, Pakistan"}]},{"given":"Shafiq ur","family":"Rehman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Engineering and Technology, Lahore 54890, Pakistan"}]},{"given":"Azqa","family":"Arif","sequence":"additional","affiliation":[{"name":"School of Natural Sciences, Department of Mathematics, National University of Sciences and Technology, Islamabad 44000, Pakistan"}]},{"given":"Saba","family":"Shafiq","sequence":"additional","affiliation":[{"name":"School of Natural Sciences, Department of Mathematics, National University of Sciences and Technology, Islamabad 44000, Pakistan"}]},{"given":"Muhammad","family":"Younas","sequence":"additional","affiliation":[{"name":"Business Administration Programme, Virtual University of Pakistan, Lahore 54000, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Sanz-Serna, J.M., and Calvo, M.P. 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Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/BFb0060019","article-title":"The effective order of Runge\u2013Kutta methods","volume":"109","author":"Butcher","year":"1969","journal-title":"Lecture Notes Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/S0168-9274(98)00043-9","article-title":"Order and effective order","volume":"28","author":"Butcher","year":"1998","journal-title":"Appl. Numer. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/S0168-9274(97)00031-7","article-title":"A generalization of singly-implicit Runge-Kutta methods","volume":"24","author":"Butcher","year":"1997","journal-title":"Appl. Numer. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1023\/A:1019176422613","article-title":"The effective order of singly-implicit Runge-Kutta methods","volume":"20","author":"Butcher","year":"1999","journal-title":"Numer. 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