{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:59:58Z","timestamp":1759838398056},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,4,16]],"date-time":"2020-04-16T00:00:00Z","timestamp":1586995200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,4,16]],"date-time":"2020-04-16T00:00:00Z","timestamp":1586995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11571302"],"award-info":[{"award-number":["11571302"]}],"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","award":["ZR2018MA024"],"award-info":[{"award-number":["ZR2018MA024"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"name":"The project of Shandong Province higher Educational Science and Technology Program","award":["Nos. J18KA247, J17KA190"],"award-info":[{"award-number":["Nos. J18KA247, J17KA190"]}]},{"name":"The foundation of innovative science and technology for youth in universities of Shandong Province","award":["2019KJI001"],"award-info":[{"award-number":["2019KJI001"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2021,3]]},"DOI":"10.1007\/s11075-020-00927-4","type":"journal-article","created":{"date-parts":[[2020,4,16]],"date-time":"2020-04-16T07:02:44Z","timestamp":1587020564000},"page":"1143-1163","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Explicit pseudo two-step exponential Runge\u2013Kutta methods for the numerical integration of first-order differential equations"],"prefix":"10.1007","volume":"86","author":[{"given":"Yonglei","family":"Fang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xianfa","family":"Hu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiyong","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,4,16]]},"reference":[{"key":"927_CR1","doi-asserted-by":"publisher","first-page":"1085","DOI":"10.1016\/0001-6160(79)90196-2","volume":"27","author":"SM Allen","year":"1979","unstructured":"Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085\u20131095 (1979)","journal-title":"Acta Metall."},{"key":"927_CR2","doi-asserted-by":"publisher","first-page":"2897","DOI":"10.1137\/110856617","volume":"50","author":"L Brugnano","year":"2012","unstructured":"Brugnano, L., Iavernaro, F., Trigiante, D.: Energy and quadratic invariants preserving integrators based upon Gauss collocation formulae. SIAM J. Numer. Anal. 50, 2897\u20132916 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"927_CR3","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1080\/00207169908804881","volume":"73","author":"NH Cong","year":"1999","unstructured":"Cong, N.H.: Explicit pseudo two-step Runge-Kutta methods for parallel computers. Int. J. Comput. Math. 73, 77\u201391 (1999)","journal-title":"Int. J. Comput. Math."},{"key":"927_CR4","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1016\/S0168-9274(01)00031-9","volume":"38","author":"NH Cong","year":"2001","unstructured":"Cong, N.H.: Explicit pseudo two-step RKN methods with stepsize control. Appl. Numer. Math. 38, 135\u2013144 (2001)","journal-title":"Appl. Numer. Math."},{"key":"927_CR5","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1023\/A:1018930732643","volume":"10","author":"NH Cong","year":"1999","unstructured":"Cong, N.H., Strehmel, K., Weiner, R.: Runge-kutta-nystr\u00f6m-type parallel block predictor-corrector methods. Adv. Comput Math. 10, 115\u2013133 (1999)","journal-title":"Adv. Comput Math."},{"key":"927_CR6","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/S0898-1221(99)00211-4","volume":"38","author":"NH Cong","year":"1999","unstructured":"Cong, N.H., Strehmel, K., Weiner, R.: A general class of explicit pseudo two-step RKN methods on parallel computers. Comput. Math. Appl. 38, 17\u201339 (1999)","journal-title":"Comput. Math. Appl."},{"key":"927_CR7","doi-asserted-by":"publisher","first-page":"430","DOI":"10.1006\/jcph.2002.6995","volume":"176","author":"S Cox","year":"2002","unstructured":"Cox, S., Matthews, P.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430\u2013455 (2002)","journal-title":"J. Comput. Phys."},{"key":"927_CR8","doi-asserted-by":"publisher","first-page":"1863","DOI":"10.1007\/s10114-018-6300-1","volume":"34","author":"YL Fang","year":"2018","unstructured":"Fang, Y.L., Liu, C.Y., Wang, B.: Efficient energy-preserving methods for general nonlinear oscillatory Hamiltonian systems. Act. Math. Sin. 34, 1863\u20131878 (2018)","journal-title":"Act. Math. Sin."},{"key":"927_CR9","doi-asserted-by":"publisher","first-page":"1235","DOI":"10.1007\/s11075-018-0619-7","volume":"81","author":"YL Fang","year":"2019","unstructured":"Fang, Y.L., Yang, Y.P., You, X.: A new family of A stable Runge Kutta methods with equation dependent coefficients for stiff problems. Numer. Algo. 81, 1235\u20131251 (2019)","journal-title":"Numer. Algo."},{"key":"927_CR10","first-page":"679","volume":"7","author":"XL Feng","year":"2013","unstructured":"Feng, X.L., Song, H.L., Tang, T., Yang, J.: Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inve. Prob. Imag. 7, 679\u2013695 (2013)","journal-title":"Inve. Prob. Imag."},{"key":"927_CR11","doi-asserted-by":"publisher","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."},{"key":"927_CR12","unstructured":"Hairer E., Lubich C., Wanner G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)"},{"key":"927_CR13","first-page":"727","volume":"18","author":"LJ Hayes","year":"1987","unstructured":"Hayes, L.J.: Gelerkin alternating direction methods for nonrectangular regions using patch approximations. SIAM J. Numer. Anal. 18, 727\u2013643 (1987)","journal-title":"SIAM J. Numer. Anal."},{"key":"927_CR14","doi-asserted-by":"publisher","first-page":"1552","DOI":"10.1137\/S1064827595295337","volume":"19","author":"M Hochbruck","year":"1998","unstructured":"Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552\u20131574 (1998)","journal-title":"SIAM J. Sci. Comput."},{"key":"927_CR15","first-page":"435","volume":"46","author":"W Kutta","year":"1901","unstructured":"Kutta, W.: Beritrag zur n\u00e4herungsweisen integration totaler differentialgleichungen. Zeitschr. f\u00fcr Math. u. Phys. 46, 435\u2013453 (1901)","journal-title":"Zeitschr. f\u00fcr Math. u. Phys."},{"key":"927_CR16","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1007\/s11075-017-0395-9","volume":"78","author":"JY Li","year":"2018","unstructured":"Li, J.Y., Deng, S., Wang, X.F.: Extended explicit pseudo two-step RKN methods for oscillatory systems $y^{\\prime \\prime }+my=f(y)$. Numer. Algo. 78, 673\u2013700 (2018)","journal-title":"Numer. Algo."},{"key":"927_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jcp.2017.10.057","volume":"356","author":"C Liu","year":"2018","unstructured":"Liu, C., Iserles, A., Wu, X.: Symmetric and arbitrarily high-order Birkhoff-Hermite time integrators and their long-time behaviour for solving nonlinear Klein-Gordon equations. J. Comput. Phys. 356, 1\u201330 (2018)","journal-title":"J. Comput. Phys."},{"key":"927_CR18","doi-asserted-by":"publisher","first-page":"352","DOI":"10.1016\/j.apnum.2020.02.020","volume":"153","author":"C Liu","year":"2020","unstructured":"Liu, C., Wu, X.: Nonlinear stability and convergence of ERKN integrators for solving nonlinear multi-frequency highly oscillatory second-order ODEs with applications to semi-linear wave equations. Appl. Numer. Math. 153, 352\u2013380 (2020)","journal-title":"Appl. Numer. Math."},{"key":"927_CR19","doi-asserted-by":"publisher","first-page":"588","DOI":"10.1016\/j.cam.2017.08.003","volume":"339","author":"C Liu","year":"2018","unstructured":"Liu, C., Wu, X., Shi, W.: New energy-preserving algorithms for nonlinear Hamiltonian wave equation equipped with Neumann boundary conditions. Appl. Math. Comput. 339, 588\u2013606 (2018)","journal-title":"Appl. Math. Comput."},{"key":"927_CR20","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1016\/j.jcp.2017.03.018","volume":"338","author":"L Mei","year":"2017","unstructured":"Mei, L., Wu, X.: Symplectic exponential Runge-Kutta methods for solving nonlinear Hamiltonian systems. J. Comput. Phys. 338, 567\u2013584 (2017)","journal-title":"J. Comput. Phys."},{"key":"927_CR21","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/BF01446807","volume":"46","author":"C Runge","year":"1895","unstructured":"Runge, C.: Ueber die numerische aufl\u00f6sung von differentialgleichungen. Math. Ann. 46, 167\u2013178 (1895)","journal-title":"Math. Ann."},{"key":"927_CR22","doi-asserted-by":"publisher","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":"927_CR23","doi-asserted-by":"publisher","first-page":"711","DOI":"10.4208\/jcm.1611-m2016-0596","volume":"35","author":"B Wang","year":"2017","unstructured":"Wang, B., Wu, X., Meng, F., Fang, Y.: Exponential Fourier collocation methods for solving first-order differential equations. J. Comput. Math. 35, 711\u2013736 (2017)","journal-title":"J. Comput. Math."},{"key":"927_CR24","doi-asserted-by":"publisher","first-page":"1335","DOI":"10.1016\/j.apnum.2012.06.019","volume":"62","author":"R Weiner","year":"2012","unstructured":"Weiner, R., El-Azab, T.: Exponential peer methods. Appl. Numer. Math. 62, 1335\u20131348 (2012)","journal-title":"Appl. Numer. Math."},{"key":"927_CR25","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1137\/040611434","volume":"43","author":"M Hochbruch","year":"2005","unstructured":"Hochbruch, M., Ostermann, A.: Explicit exponential Runge-Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43, 1069\u20131090 (2005)","journal-title":"SIAM J. Numer. Anal."},{"key":"927_CR26","doi-asserted-by":"publisher","first-page":"1955","DOI":"10.1016\/j.cpc.2010.09.006","volume":"181","author":"XY Wu","year":"2010","unstructured":"Wu, X.Y., Wang, B.: Multidimensional adapted Runge-Kutta-nystr\u00f6m methods for oscillatory systems. Comput. Phys. Commun. 181, 1955\u20131962 (2010)","journal-title":"Comput. Phys. Commun."},{"key":"927_CR27","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1016\/j.aml.2018.11.014","volume":"90","author":"YJ Wu","year":"2019","unstructured":"Wu, Y.J., Wang, B.: Symmetric and symplectic exponential integrators for nonlinear Hamiltonian systems. Appl. Math. Lett. 90, 215\u2013222 (2019)","journal-title":"Appl. Math. Lett."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00927-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-020-00927-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00927-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,16]],"date-time":"2021-04-16T00:17:55Z","timestamp":1618532275000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-020-00927-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,16]]},"references-count":27,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,3]]}},"alternative-id":["927"],"URL":"https:\/\/doi.org\/10.1007\/s11075-020-00927-4","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,16]]},"assertion":[{"value":"19 February 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 March 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 April 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}]}}