{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T16:38:04Z","timestamp":1774975084835,"version":"3.50.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,1,7]],"date-time":"2023-01-07T00:00:00Z","timestamp":1673049600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,1,7]],"date-time":"2023-01-07T00:00:00Z","timestamp":1673049600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100004377","name":"Hong Kong Polytechnic University","doi-asserted-by":"publisher","award":["4-ZZLS"],"award-info":[{"award-number":["4-ZZLS"]}],"id":[{"id":"10.13039\/501100004377","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Hong Kong Research Grant Council","award":["RFS2021-5S03"],"award-info":[{"award-number":["RFS2021-5S03"]}]},{"name":"Hong Kong Research Grant Council","award":["GRF grant 15302122"],"award-info":[{"award-number":["GRF grant 15302122"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2023,2]]},"DOI":"10.1007\/s10915-022-02094-1","type":"journal-article","created":{"date-parts":[[2023,1,7]],"date-time":"2023-01-07T09:06:47Z","timestamp":1673082407000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["An Implicit\u2013Explicit Second-Order BDF Numerical Scheme with Variable Steps for Gradient Flows"],"prefix":"10.1007","volume":"94","author":[{"given":"Dianming","family":"Hou","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0553-1404","authenticated-orcid":false,"given":"Zhonghua","family":"Qiao","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,1,7]]},"reference":[{"issue":"5","key":"2094_CR1","doi-asserted-by":"publisher","first-page":"2851","DOI":"10.1137\/120880677","volume":"51","author":"A Baskaran","year":"2013","unstructured":"Baskaran, A., Lowengrub, J.S., Wang, C., Wise, S.M.: Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 51(5), 2851\u20132873 (2013)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"2094_CR2","doi-asserted-by":"publisher","first-page":"644","DOI":"10.1007\/BF02510406","volume":"38","author":"J Becker","year":"1998","unstructured":"Becker, J.: A second order backward difference method with variable steps for a parabolic problem. BIT Numer. Math. 38(4), 644\u2013662 (1998)","journal-title":"BIT Numer. Math."},{"issue":"1","key":"2094_CR3","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/s00211-020-01112-4","volume":"145","author":"H Chen","year":"2020","unstructured":"Chen, H., Mao, J., Shen, J.: Optimal error estimates for the scalar auxiliary variable finite-element schemes for gradient flows. Numer. Math. 145(1), 167\u2013196 (2020)","journal-title":"Numer. Math."},{"key":"2094_CR4","volume":"3","author":"W Chen","year":"2019","unstructured":"Chen, W., Wang, C., Wang, X., Wise, S.M.: Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential. J. Comput. Phys. X 3, 100031 (2019)","journal-title":"J. Comput. Phys. X"},{"issue":"1","key":"2094_CR5","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1137\/18M1206084","volume":"57","author":"W Chen","year":"2019","unstructured":"Chen, W., Wang, X., Yan, Y., Zhang, Z.: A second order BDF numerical scheme with variable steps for the Cahn-Hilliard equation. SIAM J. Numer. Anal. 57(1), 495\u2013525 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"2094_CR6","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1016\/j.jcp.2019.01.014","volume":"383","author":"S Dong","year":"2019","unstructured":"Dong, S., Yang, Z., Lin, L.: A family of second-order energy-stable schemes for Cahn\u2013Hilliard type equations. J. Comput. Phys. 383, 24\u201354 (2019)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"2094_CR7","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1137\/19M1243750","volume":"63","author":"Q Du","year":"2021","unstructured":"Du, Q., Ju, L., Li, X., Qiao, Z.: Maximum bound principles for a class of semilinear parabolic equations and exponential time differencing schemes. SIAM Rev. 63(2), 317\u2013359 (2021)","journal-title":"SIAM Rev."},{"issue":"6","key":"2094_CR8","doi-asserted-by":"publisher","first-page":"1622","DOI":"10.1137\/0730084","volume":"30","author":"CM Elliott","year":"1993","unstructured":"Elliott, C.M., Stuart, A.M.: The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30(6), 1622\u20131663 (1993)","journal-title":"SIAM J. Numer. Anal."},{"key":"2094_CR9","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1557\/PROC-529-39","volume":"529","author":"DJ Eyre","year":"1998","unstructured":"Eyre, D.J.: Unconditionally gradient stable time marching the Cahn\u2013Hilliard equation. Mater. Res. Soc. Sympos. Proc. 529, 39\u201346 (1998)","journal-title":"Mater. Res. Soc. Sympos. Proc."},{"key":"2094_CR10","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1016\/j.jcp.2014.08.001","volume":"277","author":"Z Guan","year":"2014","unstructured":"Guan, Z., Lowengrub, J.S., Wang, C., Wise, S.M.: Second order convex splitting schemes for periodic nonlocal Cahn\u2013Hilliard and Allen\u2013Cahn equations. J. Comput. Phys. 277, 48\u201371 (2014)","journal-title":"J. Comput. Phys."},{"key":"2094_CR11","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1016\/j.jcp.2019.05.037","volume":"395","author":"D Hou","year":"2019","unstructured":"Hou, D., Azaiez, M., Xu, C.: A variant of scalar auxiliary variable approaches for gradient flows. J. Comput. Phys. 395, 307\u2013332 (2019)","journal-title":"J. Comput. Phys."},{"issue":"15","key":"2094_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110628","volume":"445","author":"D Hou","year":"2021","unstructured":"Hou, D., Xu, C.: Robust and stable schemes for time fractional molecular beam epitaxial growth model using SAV approach. J. Comput. Phys. 445(15), 110628 (2021)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"2094_CR13","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1007\/s10915-021-01667-w","volume":"90","author":"D Hou","year":"2022","unstructured":"Hou, D., Xu, C.: A second order energy dissipative schemes for time fractional $$L^{2}$$ gradient flows using SAV approach. J. Sci. Comput. 90(1), 25 (2022)","journal-title":"J. Sci. Comput."},{"key":"2094_CR14","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.114718","volume":"392","author":"F Huang","year":"2022","unstructured":"Huang, F., Shen, J.: A new class of implicit-explicit BDFk SAV schemes for general dissipative systems and their error analysis. Comput. Methods Appl. Mech. Engrg. 392, 114718 (2022)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"issue":"4","key":"2094_CR15","doi-asserted-by":"publisher","first-page":"A2514","DOI":"10.1137\/19M1298627","volume":"42","author":"F Huang","year":"2020","unstructured":"Huang, F., Shen, J., Yang, Z.: A highly efficient and accurate new scalar auxiliary variable approach for gradient flows. SIAM J. Sci. Comput. 42(4), A2514\u2013A2536 (2020)","journal-title":"SIAM J. Sci. Comput."},{"key":"2094_CR16","doi-asserted-by":"publisher","first-page":"1859","DOI":"10.1090\/mcom\/3262","volume":"87","author":"L Ju","year":"2021","unstructured":"Ju, L., Li, X., Qiao, Z., Zhang, H.: Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection. Math. Comput. 87, 1859\u20131885 (2021)","journal-title":"Math. Comput."},{"key":"2094_CR17","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1007\/s10915-016-0251-4","volume":"70","author":"D Li","year":"2017","unstructured":"Li, D., Qiao, Z.: On second order semi-implicit Fourier spectral methods for 2D Cahn\u2013Hilliard equations. J. Sci. Comput. 70, 301\u2013341 (2017)","journal-title":"J. Sci. Comput."},{"issue":"3","key":"2094_CR18","doi-asserted-by":"publisher","first-page":"1653","DOI":"10.1137\/140993193","volume":"54","author":"D Li","year":"2016","unstructured":"Li, D., Qiao, Z., Tang, T.: Characterizing the stabilization size for semi-implicit fourier-spectral method to phase field equations. SIAM J. Numer. Anal. 54(3), 1653\u20131681 (2016)","journal-title":"SIAM J. Numer. Anal."},{"issue":"319","key":"2094_CR19","doi-asserted-by":"publisher","first-page":"2047","DOI":"10.1090\/mcom\/3428","volume":"88","author":"X Li","year":"2019","unstructured":"Li, X., Shen, J., Rui, H.: Energy stability and convergence of SAV block-centered finite difference method for gradient flows. Math. Comput. 88(319), 2047\u20132068 (2019)","journal-title":"Math. Comput."},{"key":"2094_CR20","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1007\/s10915-022-01861-4","volume":"92","author":"H Liao","year":"2022","unstructured":"Liao, H., Ji, B., Wang, L., Zhang, Z.: Mesh-robustness of the variable steps BDF2 method for the Cahn\u2013Hilliard model. J. Sci. Comput. 92, 52 (2022)","journal-title":"J. Sci. Comput."},{"issue":"4","key":"2094_CR21","doi-asserted-by":"publisher","first-page":"2294","DOI":"10.1137\/19M1289157","volume":"58","author":"H Liao","year":"2020","unstructured":"Liao, H., Tang, T., Zhou, T.: On energy stable, maximum-principle preserving, second order BDF scheme with variable steps for the Allen\u2013Cahn equation. SIAM J. Numer. Anal. 58(4), 2294\u20132314 (2020)","journal-title":"SIAM J. Numer. Anal."},{"key":"2094_CR22","doi-asserted-by":"publisher","first-page":"1207","DOI":"10.1090\/mcom\/3585","volume":"90","author":"H Liao","year":"2020","unstructured":"Liao, H., Zhang, Z.: Analysis of adaptive BDF2 scheme for diffusion equations. Math. Comput. 90, 1207\u20131226 (2020)","journal-title":"Math. Comput."},{"issue":"3","key":"2094_CR23","doi-asserted-by":"publisher","first-page":"1395","DOI":"10.1137\/100812781","volume":"33","author":"Z Qiao","year":"2011","unstructured":"Qiao, Z., Zhang, Z., Tang, T.: An adaptive time-stepping strategy for the molecular beam epitaxy models. SIAM J. Sci. Comput. 33(3), 1395\u20131414 (2011)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"2094_CR24","doi-asserted-by":"publisher","first-page":"725","DOI":"10.1137\/0719051","volume":"19","author":"ML Roux","year":"1982","unstructured":"Roux, M.L.: Variable step size multistep methods for parabolic problems. SIAM J. Numer. Anal. 19(4), 725\u2013541 (1982)","journal-title":"SIAM J. Numer. Anal."},{"issue":"5","key":"2094_CR25","doi-asserted-by":"publisher","first-page":"2895","DOI":"10.1137\/17M1159968","volume":"56","author":"J Shen","year":"2018","unstructured":"Shen, J., Xu, J.: Convergence and error analysis for the scalar auxiliary variable(SAV) schemes to gradient flows. SIAM J. Numer. Anal. 56(5), 2895\u20132912 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"2094_CR26","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1016\/j.jcp.2017.10.021","volume":"353","author":"J Shen","year":"2018","unstructured":"Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407\u2013416 (2018)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"2094_CR27","doi-asserted-by":"publisher","first-page":"474","DOI":"10.1137\/17M1150153","volume":"61","author":"J Shen","year":"2019","unstructured":"Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474\u2013506 (2019)","journal-title":"SIAM Rev."},{"issue":"4","key":"2094_CR28","doi-asserted-by":"publisher","first-page":"1669","DOI":"10.3934\/dcds.2010.28.1669","volume":"28","author":"J Shen","year":"2010","unstructured":"Shen, J., Yang, X.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Conti. Dyn. Syst. Ser. A 28(4), 1669\u20131691 (2010)","journal-title":"Discrete Conti. Dyn. Syst. Ser. A"},{"key":"2094_CR29","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0645-3","volume-title":"Infinite-Dimensional Dynamical Systems in Mechanics and Physics","author":"R Temam","year":"1997","unstructured":"Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin (1997)"},{"issue":"6","key":"2094_CR30","first-page":"838","volume":"26","author":"D Wang","year":"2008","unstructured":"Wang, D., Steven, J.R.: Variable step-size implicit-explicit linear mulstep methods for time-dependent partial differential equations. J. Comput. Math. 26(6), 838\u2013855 (2008)","journal-title":"J. Comput. Math."},{"key":"2094_CR31","unstructured":"Wei, Y., Zhang, J., Zhao, C., Zhao,Y.: A unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn\u2013Hilliard equation on generalized SAV approach. arXiv:2211.02018, (2022)"},{"issue":"4","key":"2094_CR32","doi-asserted-by":"publisher","first-page":"1759","DOI":"10.1137\/050628143","volume":"44","author":"C Xu","year":"2006","unstructured":"Xu, C., Tang, T.: Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44(4), 1759\u20131779 (2006)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"2094_CR33","doi-asserted-by":"publisher","first-page":"572","DOI":"10.4208\/cicp.OA-2016-0197","volume":"23","author":"Y Yan","year":"2018","unstructured":"Yan, Y., Chen, W., Wang, C., Wise, S.M.: A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation. Commun. Comput. Phys. 23(2), 572\u2013602 (2018)","journal-title":"Commun. Comput. Phys."},{"key":"2094_CR34","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1016\/j.jcp.2016.09.029","volume":"327","author":"X Yang","year":"2016","unstructured":"Yang, X.: Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294\u2013316 (2016)","journal-title":"J. Comput. Phys."},{"issue":"5","key":"2094_CR35","first-page":"377","volume":"42","author":"J Zhang","year":"2022","unstructured":"Zhang, J., Zhao, C.: Sharp error estimate of BDF2 scheme with variable time steps for molecular beam expitaxial models without slop selection. J. Math. 42(5), 377\u2013401 (2022)","journal-title":"J. Math."},{"key":"2094_CR36","doi-asserted-by":"crossref","unstructured":"Zhao, C., Liu, N., Ma, Y., Zhang, J.: Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equations. arXiv:2201.06008, (2022)","DOI":"10.4310\/CMS.2023.v21.n3.a7"},{"issue":"3","key":"2094_CR37","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1002\/nme.5372","volume":"110","author":"J Zhao","year":"2017","unstructured":"Zhao, J., Wang, Q., Yang, X.: Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach. Int. J. Numer. Meth. Eng. 110(3), 279\u2013300 (2017)","journal-title":"Int. J. Numer. Meth. Eng."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-022-02094-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-022-02094-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-022-02094-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,20]],"date-time":"2023-03-20T03:27:50Z","timestamp":1679282870000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-022-02094-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,7]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["2094"],"URL":"https:\/\/doi.org\/10.1007\/s10915-022-02094-1","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,7]]},"assertion":[{"value":"4 July 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 November 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 December 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have not disclosed any competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interests"}}],"article-number":"39"}}