{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T16:02:38Z","timestamp":1759939358113,"version":"build-2065373602"},"reference-count":45,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2025,8,15]],"date-time":"2025-08-15T00:00:00Z","timestamp":1755216000000},"content-version":"vor","delay-in-days":1475,"URL":"https:\/\/linproxy.fan.workers.dev:443\/http\/www.elsevier.com\/open-access\/userlicense\/1.0\/"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Computational and Applied Mathematics"],"published-print":{"date-parts":[[2021,8]]},"DOI":"10.1016\/j.cam.2020.113312","type":"journal-article","created":{"date-parts":[[2021,2,21]],"date-time":"2021-02-21T04:16:09Z","timestamp":1613880969000},"page":"113312","update-policy":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":8,"special_numbering":"C","title":["Two-frequency trigonometrically-fitted and symmetric linear multi-step methods for second-order oscillators"],"prefix":"10.1016","volume":"392","author":[{"given":"Yonglei","family":"Fang","sequence":"first","affiliation":[]},{"given":"Ting","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Xiong","family":"You","sequence":"additional","affiliation":[]},{"given":"Juan","family":"Zheng","sequence":"additional","affiliation":[]},{"given":"Bin","family":"Wang","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/j.cam.2020.113312_b1","doi-asserted-by":"crossref","first-page":"1344","DOI":"10.1007\/s11182-006-0264-9","article-title":"Special perturbation theory methods in celestial mechanics, I. Principles for the construction and substantiation of the application","volume":"49","author":"Avdyushev","year":"2006","journal-title":"Russian Phys. J."},{"year":"1993","series-title":"Solving Ordinary Differential Equations I: Nonstiff Problems","author":"Hairer","key":"10.1016\/j.cam.2020.113312_b2"},{"year":"1982","series-title":"Mechanics","author":"Landau","key":"10.1016\/j.cam.2020.113312_b3"},{"key":"10.1016\/j.cam.2020.113312_b4","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/S0010-4655(99)00365-3","article-title":"Exponentially fitted Runge\u2013Kutta methods","volume":"123","author":"Vanden\u00a0Berghe","year":"1999","journal-title":"Comput. Phys. Comm."},{"key":"10.1016\/j.cam.2020.113312_b5","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/S0010-4655(01)00285-5","article-title":"An exponentially-fitted high order method for long-term integration of periodic initial-value problems","volume":"140","author":"Simos","year":"2001","journal-title":"Comput. Phys. Comm."},{"key":"10.1016\/j.cam.2020.113312_b6","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/S0010-4655(00)00080-1","article-title":"An embedded exponentially-fitted Runge\u2013Kutta method for the numerical solution of the Schr\u00f6dinger equation and related periodic initial-value problems","volume":"131","author":"Avdelas","year":"2000","journal-title":"Comput. Phys. Comm."},{"key":"10.1016\/j.cam.2020.113312_b7","doi-asserted-by":"crossref","first-page":"1035","DOI":"10.1142\/S0129183101002292","article-title":"A symmetric high order method with minimal phase-lag for the numerical solution of the Schr\u00f6dinger equation","volume":"12","author":"Simos","year":"2001","journal-title":"Internat. J. Modern Phys. C"},{"key":"10.1016\/j.cam.2020.113312_b8","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.cam.2006.04.033","article-title":"A fourth-order Runge\u2013Kutta method based on BDF-type Chebyshev approximations","volume":"204","author":"Ramos","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2020.113312_b9","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1093\/imanum\/drl040","article-title":"A family of A-stable Runge\u2013Kutta collocation methods of higher order for initial-value problems","volume":"27","author":"Vigo-Aguiar","year":"2007","journal-title":"IMA J. Numer. Aanl."},{"key":"10.1016\/j.cam.2020.113312_b10","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1016\/j.mcm.2005.09.011","article-title":"Variable stepsize St\u00f6rmer-Cowell methods","volume":"42","author":"Ramos","year":"2005","journal-title":"Math. Comput. Modelling"},{"key":"10.1016\/j.cam.2020.113312_b11","doi-asserted-by":"crossref","first-page":"1656","DOI":"10.1086\/322107","article-title":"An exponentially fitted and trigonometrically fitted method for the numerical solution of orbital problems","volume":"122","author":"Vigo-Aguiar","year":"2001","journal-title":"Astron. J."},{"key":"10.1016\/j.cam.2020.113312_b12","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1063\/1.168717","article-title":"Higher-order variable step algorithms adapted to the accurate numerical integration of perturbed oscillators","volume":"12","author":"Vigo-Aguiar","year":"1998","journal-title":"Comput. Phys."},{"key":"10.1016\/j.cam.2020.113312_b13","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.cam.2014.09.008","article-title":"On the choice of the frequency in trigonometrically-fitted methods for periodic problems","volume":"277","author":"Vigo-Aguiar","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2020.113312_b14","doi-asserted-by":"crossref","first-page":"1856","DOI":"10.1016\/j.cam.2010.07.004","article-title":"A numerical ODE solver that preserves the fixed points and their stability","volume":"235","author":"Vigo-Aguiar","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2020.113312_b15","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1016\/j.cam.2015.12.005","article-title":"A first approach in solving initial-value problems in ODEs by elliptic fitting methods","volume":"318","author":"Vigo-Aguiar","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2020.113312_b16","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.apnum.2017.04.008","article-title":"Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations","volume":"119","author":"Wang","year":"2017","journal-title":"Appl. Numer. Math."},{"key":"10.1016\/j.cam.2020.113312_b17","doi-asserted-by":"crossref","first-page":"711","DOI":"10.4208\/jcm.1611-m2016-0596","article-title":"Exponential Fourier collocation methods for solving first-order differential equations","volume":"35","author":"Wang","year":"2017","journal-title":"J. Comput. Math."},{"key":"10.1016\/j.cam.2020.113312_b18","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.cam.2016.09.017","article-title":"Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second order differential equations","volume":"313","author":"Wang","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2020.113312_b19","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/s10092-016-0179-y","article-title":"Sixth order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear hamiltonian equations","volume":"54","author":"Wang","year":"2017","journal-title":"Calcolo"},{"key":"10.1016\/j.cam.2020.113312_b20","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.aml.2017.04.026","article-title":"The boundness of the operator-valued functions for multidimensional nonlinear wave equations with applications","volume":"74","author":"Liu","year":"2017","journal-title":"Appl. Math. Lett."},{"key":"10.1016\/j.cam.2020.113312_b21","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/s10208-014-9241-9","article-title":"Arbitrary-order trigonometric Fourier collocation methods for multi-frequency oscillatory systems","volume":"16","author":"Wang","year":"2016","journal-title":"Found. Comput. Math."},{"key":"10.1016\/j.cam.2020.113312_b22","doi-asserted-by":"crossref","first-page":"1998","DOI":"10.1002\/mma.4727","article-title":"Triangular splitting implementation of RKN-type Fourier collocation methods for second-order differential equations","volume":"41","author":"Wang","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"10.1016\/j.cam.2020.113312_b23","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/BF01931689","article-title":"On accuracy and unconditional stability of linear multistep methods for second order differential equations","volume":"18","author":"Dahlquist","year":"1978","journal-title":"BIT"},{"key":"10.1016\/j.cam.2020.113312_b24","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1093\/imamat\/18.2.189","article-title":"Symmetric multistep methods for periodic initial value problems","volume":"18","author":"Lambert","year":"1976","journal-title":"J. Inst. Math. Appl."},{"year":"2004","series-title":"Exponential Fitting","author":"Ixaru","key":"10.1016\/j.cam.2020.113312_b25"},{"key":"10.1016\/j.cam.2020.113312_b26","doi-asserted-by":"crossref","first-page":"1863","DOI":"10.1007\/s10114-018-6300-1","article-title":"Efficient energy-preserving methods for general nonlinear oscillatory Hamiltonian system","volume":"34","author":"Fang","year":"2018","journal-title":"Act. Math. Sin."},{"key":"10.1016\/j.cam.2020.113312_b27","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1016\/j.cam.2016.09.016","article-title":"Revised trigonometrically fitted two step hybrid methods with equation dependent coefficients for highly oscillatory problems","volume":"318","author":"Fang","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2020.113312_b28","doi-asserted-by":"crossref","unstructured":"Y.L. Fang, X.F. Hu, Explicit pseudo two-step exponential Runge\u2013Kutta methods for the numerical integration of first-order differential equations, Numer. Algorithms https:\/\/linproxy.fan.workers.dev:443\/http\/dx.doi.org\/10.1007\/s11075-020-00927-4.","DOI":"10.1007\/s11075-020-00927-4"},{"key":"10.1016\/j.cam.2020.113312_b29","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1016\/j.cpc.2011.04.001","article-title":"Trigonometrically-fitted Scheifele two-step methods for perturbed oscillators","volume":"182","author":"You","year":"2011","journal-title":"Comput. Phys. Comm."},{"key":"10.1016\/j.cam.2020.113312_b30","doi-asserted-by":"crossref","first-page":"1777","DOI":"10.1016\/j.cpc.2009.05.010","article-title":"Extended RKN-type methods for numerical integration of perturbed oscillators","volume":"180","author":"Yang","year":"2009","journal-title":"Comput. Phys. Comm."},{"key":"10.1016\/j.cam.2020.113312_b31","doi-asserted-by":"crossref","first-page":"2486","DOI":"10.1016\/j.cpc.2011.07.007","article-title":"Two-step extended RKN methods for oscillatory systems","volume":"182","author":"Li","year":"2011","journal-title":"Comput. Phys. Comm."},{"key":"10.1016\/j.cam.2020.113312_b32","first-page":"6241","article-title":"New explicit adapted numerov methods for second-order oscillatory differential equations","volume":"219","author":"You","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.cam.2020.113312_b33","first-page":"597","article-title":"A novel family of P-stable symmetric extended linear multistep methods for oscillators","volume":"249","author":"You","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.cam.2020.113312_b34","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/BF01395931","article-title":"Chebyshevian multistep methods for ordinary differential equations","volume":"19","author":"Lyche","year":"1972","journal-title":"Numer. Math."},{"key":"10.1016\/j.cam.2020.113312_b35","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.cam.2004.03.020","article-title":"Two-frequency-dependent Gauss quadrature rules","volume":"174","author":"Kim","year":"2005","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2020.113312_b36","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/j.cpc.2006.03.004","article-title":"Trigonometrically-fitted method for a periodic initial value problem with two frequencies","volume":"175","author":"Wang","year":"2006","journal-title":"Comput. Phys. Comm."},{"key":"10.1016\/j.cam.2020.113312_b37","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1016\/j.cpc.2008.07.013","article-title":"Trigonometrically fitted explicit Numerov-type method for periodic IVPs with two frequencies","volume":"179","author":"Fang","year":"2008","journal-title":"Comput. Phys. Comm."},{"year":"1973","series-title":"Computational Methods in Ordinary Differential Equations","author":"Lambert","key":"10.1016\/j.cam.2020.113312_b38"},{"key":"10.1016\/j.cam.2020.113312_b39","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1093\/imanum\/16.2.179","article-title":"P-stability and exponential-fitting methods for y\u2032\u2032=f(x,y)","volume":"16","author":"Coleman","year":"1996","journal-title":"IMA J. Numer. Anal."},{"key":"10.1016\/j.cam.2020.113312_b40","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/S0377-0427(96)00156-2","article-title":"A finite-difference method for the numerical solution of the Schr\u00f6dinger equation","volume":"79","author":"Simos","year":"1997","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2020.113312_b41","doi-asserted-by":"crossref","first-page":"1626","DOI":"10.1016\/j.cpc.2011.04.011","article-title":"A symmetric eight-step predictor\u2013corrector method for the numerical solution of the radial Schr\u00f6dinger equation and related IVPs with oscillating solutions","volume":"182","author":"Panopoulos","year":"2011","journal-title":"Comput. Phys. Comm."},{"key":"10.1016\/j.cam.2020.113312_b42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S1384-1076(01)00084-7","article-title":"Exponentially-fitted and trigonometrically-fitted methods for long-term integration of orbital problems","volume":"7","author":"Simos","year":"2002","journal-title":"New Astron."},{"key":"10.1016\/j.cam.2020.113312_b43","doi-asserted-by":"crossref","first-page":"1694","DOI":"10.1086\/115629","article-title":"Symmetric multistep methods for the numerical integration of planetary orbits","volume":"100","author":"Quinlan","year":"1990","journal-title":"Astron. J."},{"key":"10.1016\/j.cam.2020.113312_b44","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1016\/j.cpc.2013.10.005","article-title":"A new phase-fitted eight-step symmetric embedded predictor\u2013corrector method (EPCM) for orbital problems and related IVPs with oscillating solutions","volume":"185","author":"Panopoulos","year":"2014","journal-title":"Comput. Phys. Comm."},{"key":"10.1016\/j.cam.2020.113312_b45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0377-0427(90)90001-G","article-title":"High-order P-stable multistep methods","volume":"30","author":"Franco","year":"1990","journal-title":"J. Comput. Appl. Math."}],"container-title":["Journal of Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/api.elsevier.com\/content\/article\/PII:S0377042720306038?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/api.elsevier.com\/content\/article\/PII:S0377042720306038?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:18:15Z","timestamp":1759835895000},"score":1,"resource":{"primary":{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/linkinghub.elsevier.com\/retrieve\/pii\/S0377042720306038"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8]]},"references-count":45,"alternative-id":["S0377042720306038"],"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1016\/j.cam.2020.113312","relation":{},"ISSN":["0377-0427"],"issn-type":[{"type":"print","value":"0377-0427"}],"subject":[],"published":{"date-parts":[[2021,8]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Two-frequency trigonometrically-fitted and symmetric linear multi-step methods for second-order oscillators","name":"articletitle","label":"Article Title"},{"value":"Journal of Computational and Applied Mathematics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1016\/j.cam.2020.113312","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2021 Elsevier B.V.","name":"copyright","label":"Copyright"}],"article-number":"113312"}}