{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T06:17:11Z","timestamp":1774937831610,"version":"3.50.1"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2016,11,16]],"date-time":"2016-11-16T00:00:00Z","timestamp":1479254400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["11271186"],"award-info":[{"award-number":["11271186"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["11671200"],"award-info":[{"award-number":["11671200"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2017,8]]},"DOI":"10.1007\/s11075-016-0225-5","type":"journal-article","created":{"date-parts":[[2016,11,17]],"date-time":"2016-11-17T04:15:26Z","timestamp":1479356126000},"page":"909-935","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["An improved tri-coloured rooted-tree theory and order conditions for ERKN methods for general multi-frequency oscillatory systems"],"prefix":"10.1007","volume":"75","author":[{"given":"Xianyang","family":"Zeng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongli","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinyuan","family":"Wu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,11,16]]},"reference":[{"key":"225_CR1","unstructured":"Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. 2nd edn. Springer, Berlin (2006)"},{"key":"225_CR2","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-35338-3","volume-title":"Structure-preserving algorithms for oscillatory differential equations","author":"X Wu","year":"2013","unstructured":"Wu, X., You, X., Wang, B.: Structure-preserving algorithms for oscillatory differential equations. Springer, Berlin (2013)"},{"key":"225_CR3","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-48156-1","volume-title":"Structure-preserving algorithms for oscillatory differential equations II","author":"X Wu","year":"2015","unstructured":"Wu, X., Liu, K., Shi, W.: Structure-preserving algorithms for oscillatory differential equations II. Springer, Berlin (2015)"},{"key":"225_CR4","doi-asserted-by":"crossref","first-page":"7671","DOI":"10.1016\/j.jcp.2012.07.004","volume":"231","author":"W Shi","year":"2012","unstructured":"Shi, W., Wu, X., Xia, J.: Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations. J. Comput. Phys. 231, 7671\u20137694 (2012)","journal-title":"J. Comput. Phys."},{"key":"225_CR5","doi-asserted-by":"crossref","first-page":"6505","DOI":"10.1016\/j.apm.2013.01.029","volume":"37","author":"X Wu","year":"2013","unstructured":"Wu, X., Wang, B., Shi, W.: Effective integrators for nonlinear second-order oscillatory systems with a time-dependent frequency matrix. Appl. Math. Model. 37, 6505\u20136518 (2013)","journal-title":"Appl. Math. Model."},{"key":"225_CR6","doi-asserted-by":"crossref","first-page":"2327","DOI":"10.1016\/j.apm.2012.05.021","volume":"37","author":"X Wu","year":"2013","unstructured":"Wu, X., Wang, B., Liu, K., Zhao, H.: ERKN methods for long-term integration of multidimensional orbital problems. Appl. Math. Model. 37, 2327\u20132336 (2013)","journal-title":"Appl. Math. Model."},{"key":"225_CR7","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.jmaa.2016.03.038","volume":"440","author":"C Liu","year":"2016","unstructured":"Liu, C., Wu, X.: An energy-preserving and symmetric scheme for nonlinear Hamiltonian wave equations. J. Math. Anal. Appl. 440, 167\u2013182 (2016)","journal-title":"J. Math. Anal. Appl."},{"key":"225_CR8","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.cam.2016.01.022","volume":"301","author":"X Wu","year":"2016","unstructured":"Wu, X., Liu, C., Mei, L.: A new framework for solving partial differential equations using semi-analytical explicit RK(N)-type integrators. J. Comput. Appl. Math. 301, 74\u201390 (2016)","journal-title":"J. Comput. Appl. Math."},{"key":"225_CR9","doi-asserted-by":"crossref","first-page":"1777","DOI":"10.1016\/j.cpc.2009.05.010","volume":"180","author":"H Yang","year":"2009","unstructured":"Yang, H., Wu, X., You, X., Fang, Y.: Extended RKN-type methods for numerical integration of perturbed oscillators. Comput. Phys. Commun. 180, 1777\u20131794 (2009)","journal-title":"Comput. Phys. Commun."},{"key":"225_CR10","doi-asserted-by":"crossref","first-page":"1873","DOI":"10.1016\/j.cpc.2010.07.046","volume":"181","author":"X Wu","year":"2010","unstructured":"Wu, X., You, X., Shi, W., Wang, B.: ERKN integrators for systems of oscillatory second-order differential equations. Comput. Phys. Commun. 181, 1873\u20131887 (2010)","journal-title":"Comput. Phys. Commun."},{"key":"225_CR11","doi-asserted-by":"crossref","first-page":"2486","DOI":"10.1016\/j.cpc.2011.07.007","volume":"182","author":"J Li","year":"2011","unstructured":"Li, J., Wang, B., You, X., Wu, X.: Two-step extended RKN methods for oscillatory systems. Comput. Phys. Commun. 182, 2486\u20132507 (2011)","journal-title":"Comput. Phys. Commun."},{"key":"225_CR12","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1007\/s11075-013-9761-4","volume":"65","author":"J Li","year":"2014","unstructured":"Li, J., Wu, X.: Error analysis of explicit TSERKN methods for highly oscillatory systems. Numer. Algorithms 65, 465\u2013483 (2014)","journal-title":"Numer. Algorithms"},{"key":"225_CR13","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/s11075-012-9583-9","volume":"62","author":"J Li","year":"2013","unstructured":"Li, J., Wu, X.: Adapted Falkner-type methods solving oscillatory second-order differential equations. Numer. Algorithms 62, 355\u2013381 (2013)","journal-title":"Numer. Algorithms"},{"key":"225_CR14","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1016\/j.mcm.2012.09.014","volume":"57","author":"B Wang","year":"2013","unstructured":"Wang, B., Wu, X., Zhao, H.: Novel improved multidimensional Str\u00f6mer-Verlet formulas with applications to four aspects in scientific computation. Math. Comput. Model. 57, 857\u2013 872 (2013)","journal-title":"Math. Comput. Model."},{"key":"225_CR15","doi-asserted-by":"crossref","first-page":"1185","DOI":"10.1016\/j.physleta.2012.02.040","volume":"376","author":"B Wang","year":"2012","unstructured":"Wang, B., Wu, X.: A new high precision energy-preserving integrator for system of oscillatory second-order differential equations. Phys. Lett. A. 376, 1185\u20131190 (2012)","journal-title":"Phys. Lett. A."},{"key":"225_CR16","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1016\/j.jcp.2012.10.015","volume":"235","author":"X Wu","year":"2013","unstructured":"Wu, X., Wang, B., Shi, W.: Efficient energy-preserving integrators for oscillatory Hamiltonian systems. J. Comput. Phys. 235, 587\u2013605 (2013)","journal-title":"J. Comput. Phys."},{"key":"225_CR17","unstructured":"Wu, X., Wang, B., Xia, J.: Extended symplectic Runge-Kutta-Nystr\u00f6m integrators for separable Hamiltonian systems. In: Proceedings of the 2010 International Conference on Computational and Mathematical Methods in Science and Engineering, Vol. III, pp. 1016\u20131020. Spain (2010)"},{"key":"225_CR18","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1007\/s10543-012-0379-z","volume":"52","author":"X Wu","year":"2012","unstructured":"Wu, X., Wang, B., Xia, J.: Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nystr\u00f6m methods. BIT Numer. Math. 52, 773\u2013795 (2012)","journal-title":"BIT Numer. Math."},{"key":"225_CR19","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1007\/s11075-013-9811-y","volume":"65","author":"B Wang","year":"2014","unstructured":"Wang, B., Wu, X.: A highly accurate explicit symplectic ERKN method for multi-frequency and multidimensional oscillatory Hamiltonian systems. Numer. Algorithms 65, 705\u2013721 (2014)","journal-title":"Numer. Algorithms"},{"key":"225_CR20","doi-asserted-by":"crossref","first-page":"1375","DOI":"10.1016\/j.apnum.2007.08.001","volume":"58","author":"H Yang","year":"2008","unstructured":"Yang, H., Wu, X.: Trigonometrically-fitted ARKN methods for perturbed oscillators. Appl. Numer. Math. 58, 1375\u20131395 (2008)","journal-title":"Appl. Numer. Math."},{"key":"225_CR21","doi-asserted-by":"crossref","first-page":"2250","DOI":"10.1016\/j.cpc.2009.07.011","volume":"180","author":"X Wu","year":"2009","unstructured":"Wu, X., You, X., Xia, J.: Order conditions for ARKN methods solving oscillatory system. Comput. Phys. Commun. 180, 2250\u20132257 (2009)","journal-title":"Comput. Phys. Commun."},{"key":"225_CR22","doi-asserted-by":"crossref","first-page":"6331","DOI":"10.1016\/j.apm.2012.01.053","volume":"36","author":"X Wu","year":"2012","unstructured":"Wu, X.: A note on stability of multidimensional adapted Runge-Kutta-Nystr\u00f6m methods for oscillatory systems. Appl. Math. Model. 36, 6331\u20136337 (2012)","journal-title":"Appl. Math. Model."},{"key":"225_CR23","doi-asserted-by":"crossref","first-page":"2408","DOI":"10.1016\/j.cpc.2013.06.001","volume":"184","author":"W Shi","year":"2013","unstructured":"Shi, W., Wu, X.: A note on symplectic and symmetric ARKN methods. Comput. Phys. Comm. 184, 2408\u20132411 (2013)","journal-title":"Comput. Phys. Comm."},{"key":"225_CR24","doi-asserted-by":"crossref","first-page":"1999","DOI":"10.1016\/j.cpc.2014.04.001","volume":"185","author":"K Liu","year":"2014","unstructured":"Liu, K., Wu, X.: Multidimensional ARKN methods for general oscillatory second-order initial value problems. Comput. Phys. Comm. 185, 1999\u20132007 (2014)","journal-title":"Comput. Phys. Comm."},{"key":"225_CR25","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s11075-013-9728-5","volume":"66","author":"X You","year":"2014","unstructured":"You, X., Zhao, J., Yang, H., Fang, Y., Wu, X.: Order conditions for RKN methods solving general second-order oscillatory systems. Numer. Algorithms 66, 147\u2013176 (2014)","journal-title":"Numer. Algorithms"},{"key":"225_CR26","doi-asserted-by":"crossref","first-page":"2841","DOI":"10.1016\/j.cpc.2014.07.002","volume":"185","author":"H Yang","year":"2014","unstructured":"Yang, H., Zeng, X., Wu, X., Ru, Z.: A simplified Nystr\u00f6m-tree theory for extended Runge-Kutta-Nystr\u00f6m integrators solving multi-frequency oscillatory systems. Comput. Phys. Commun. 185, 2841\u20132850 (2014)","journal-title":"Comput. Phys. Commun."},{"key":"225_CR27","unstructured":"Boik, R.J.: Lecture notes: statistics 550 spring 2006, \n                        http:\/\/www.math.montana.edu\/~rjboik\/classes\/550\/notes.550.06.pdf\n                        \n                    , 33\u201335"},{"key":"225_CR28","volume-title":"Solving ordinary differential equations I, nonstiff problems, end edn., Springer series in Computational Mathematics","author":"E Hairer","year":"1993","unstructured":"Hairer, E., N\u00f8rsett, S.P., Wanner, G.: Solving ordinary differential equations I, nonstiff problems, end edn., Springer series in Computational Mathematics. Springer, Berlin (1993)"},{"key":"225_CR29","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1090\/S0025-5718-1972-0305608-0","volume":"26","author":"JC Butcher","year":"1972","unstructured":"Butcher, J.C.: An algebraic theory of integration methods. Math. Comput. 26, 79\u2013106 (1972)","journal-title":"Math. Comput."},{"key":"225_CR30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02268387","volume":"13","author":"E Hairer","year":"1974","unstructured":"Hairer, E., Wanner, G.: On the Butcher group and general multi-value methods. Computing 13, 1\u201315 (1974)","journal-title":"Computing"},{"key":"225_CR31","volume-title":"A First Course in Partial Differential Equations with Complex Variables and Transform Methods","author":"HF Weinberger","year":"1965","unstructured":"Weinberger, H.F.: A First Course in Partial Differential Equations with Complex Variables and Transform Methods. Dover Publications Inc., New York (1965)"},{"key":"225_CR32","doi-asserted-by":"crossref","first-page":"1040","DOI":"10.1016\/j.apnum.2005.09.005","volume":"56","author":"JM Franco","year":"2006","unstructured":"Franco, J.M.: New methods for oscillatory systems based on ARKN methods. Appl. Numer. Math. 56, 1040\u20131053 (2006)","journal-title":"Appl. Numer. Math."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-016-0225-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-016-0225-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-016-0225-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,7,21]],"date-time":"2017-07-21T06:08:21Z","timestamp":1500617301000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-016-0225-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11,16]]},"references-count":32,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2017,8]]}},"alternative-id":["225"],"URL":"https:\/\/doi.org\/10.1007\/s11075-016-0225-5","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,11,16]]}}}