{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T01:53:22Z","timestamp":1771034002901,"version":"3.50.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,4,16]],"date-time":"2025-04-16T00:00:00Z","timestamp":1744761600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,16]],"date-time":"2025-04-16T00:00:00Z","timestamp":1744761600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100016073","name":"Key Technologies Research and Development Program of Anhui Province","doi-asserted-by":"publisher","award":["2020YFA0709803"],"award-info":[{"award-number":["2020YFA0709803"]}],"id":[{"id":"10.13039\/100016073","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071481"],"award-info":[{"award-number":["12071481"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271523"],"award-info":[{"award-number":["12271523"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100017548","name":"National Defense Science and Technology Innovation Fund of the Chinese Academy of Sciences","doi-asserted-by":"publisher","award":["2021-JCJQ-JJ-0538"],"award-info":[{"award-number":["2021-JCJQ-JJ-0538"]}],"id":[{"id":"10.13039\/501100017548","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science and Technology Innovation Program of Hunan Province","award":["2021RC3082"],"award-info":[{"award-number":["2021RC3082"]}]},{"name":"Science and Technology Innovation Program of Hunan Province","award":["2022RC1192"],"award-info":[{"award-number":["2022RC1192"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2025,6]]},"DOI":"10.1007\/s10915-025-02839-8","type":"journal-article","created":{"date-parts":[[2025,4,16]],"date-time":"2025-04-16T06:29:42Z","timestamp":1744784982000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Convergence and Energy Stability Analysis for a Second-Order Scheme of the Swift\u2013Hohenberg Equation"],"prefix":"10.1007","volume":"103","author":[{"given":"Jingwei","family":"Sun","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haifeng","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xu","family":"Qian","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Songhe","family":"Song","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,16]]},"reference":[{"key":"2839_CR1","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/S0168-9274(97)00056-1","volume":"25","author":"UM Ascher","year":"1997","unstructured":"Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge\u2013Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151\u2013167 (1997)","journal-title":"Appl. Numer. Math."},{"key":"2839_CR2","doi-asserted-by":"publisher","first-page":"A906","DOI":"10.1137\/23M1562056","volume":"46","author":"M Caliari","year":"2024","unstructured":"Caliari, M., Cassini, F., Einkemmer, L., Ostermann, A.: Accelerating exponential integrators to efficiently solve semilinear advection-diffusion-reaction equations. SIAM J. Sci. Comput. 46, A906\u2013A928 (2024)","journal-title":"SIAM J. Sci. Comput."},{"key":"2839_CR3","doi-asserted-by":"publisher","first-page":"546","DOI":"10.1007\/s10915-011-9559-2","volume":"52","author":"W Chen","year":"2012","unstructured":"Chen, W., Conde, S., Wang, C., Wang, X., Wise, S.M.: A linear energy stable scheme for a thin film model without slope selection. J. Sci. Comput. 52, 546\u2013562 (2012)","journal-title":"J. Sci. Comput."},{"key":"2839_CR4","doi-asserted-by":"publisher","first-page":"574","DOI":"10.1007\/s10915-013-9774-0","volume":"59","author":"W Chen","year":"2014","unstructured":"Chen, W., Wang, C., Wang, X., Wise, S.M.: A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection. J. Sci. Comput. 59, 574\u2013601 (2014)","journal-title":"J. Sci. Comput."},{"key":"2839_CR5","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1051\/m2an\/2019054","volume":"54","author":"W Chen","year":"2020","unstructured":"Chen, W., Li, W., Luo, Z., Wang, C., Wang, X.: A stabilized second order exponential time differencing multistep method for thin film growth model without slope selection. ESAIM Math. Modeling Numer. Anal. 54, 727\u2013750 (2020)","journal-title":"ESAIM Math. Modeling Numer. Anal."},{"key":"2839_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40687-020-00212-9","volume":"7","author":"W Chen","year":"2020","unstructured":"Chen, W., Li, W., Wang, C., Wang, S., Wang, X.: Energy stable higher-order linear ETD multi-step methods for gradient flows: application to thin film epitaxy. Res. Math. Sci. 7, 1\u201327 (2020)","journal-title":"Res. Math. Sci."},{"key":"2839_CR7","doi-asserted-by":"publisher","first-page":"6241","DOI":"10.1016\/j.jcp.2008.03.012","volume":"227","author":"M Cheng","year":"2008","unstructured":"Cheng, M., Warren, J.A.: An efficient algorithm for solving the phase field crystal model. J. Comput. Phys. 227, 6241\u20136248 (2008)","journal-title":"J. Comput. Phys."},{"key":"2839_CR8","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1090\/S0025-5718-10-02365-3","volume":"80","author":"N Condette","year":"2011","unstructured":"Condette, N., Melcher, C., Suli, E.: Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth. Math. Comput. 80, 205\u2013223 (2011)","journal-title":"Math. Comput."},{"key":"2839_CR9","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1103\/RevModPhys.65.851","volume":"65","author":"MC Cross","year":"1993","unstructured":"Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851 (1993)","journal-title":"Rev. Mod. Phys."},{"key":"2839_CR10","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1007\/s10915-024-02490-9","volume":"99","author":"M Cui","year":"2024","unstructured":"Cui, M., Niu, Y., Xu, Z.: A second order exponential time differencing multi-step energy stable scheme for SH equation with quadratic-cubic nonlinear term. J. Sci. Comput. 99, 26 (2024)","journal-title":"J. Sci. Comput."},{"key":"2839_CR11","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1016\/j.cma.2015.09.018","volume":"298","author":"M Dehghan","year":"2016","unstructured":"Dehghan, M., Mohammadi, V.: The numerical simulation of the phase field crystal (PFC) and modified phase field crystal (MPFC) models via global and local meshless methods. Comput. Methods Appl. Mech. Eng. 298, 453\u2013484 (2016)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2839_CR12","first-page":"59","volume":"3","author":"X Feng","year":"2013","unstructured":"Feng, X., Tang, T., Yang, J.: Stabilized Crank-Nicolson\/Adams-Bashforth schemes for phase field models. East Asain J. Appl. Math. 3, 59\u201380 (2013)","journal-title":"East Asain J. Appl. Math."},{"key":"2839_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.110943","volume":"454","author":"Z Fu","year":"2022","unstructured":"Fu, Z., Yang, J.: Energy-decreasing exponential time differencing Runge-Kutta methods for phase-field models. J. Comput. Phys. 454, 110943 (2022)","journal-title":"J. Comput. Phys."},{"key":"2839_CR14","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3950","author":"Z Fu","year":"2024","unstructured":"Fu, Z., Tang, T., Yang, J.: Energy diminishing implicit-explicit Runge-Kutta methods for gradient flows. Math. Comput. (2024). https:\/\/doi.org\/10.1090\/mcom\/3950","journal-title":"Math. Comput."},{"key":"2839_CR15","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1007\/s10915-012-9621-8","volume":"53","author":"S Gottlieb","year":"2012","unstructured":"Gottlieb, S., Wang, C.: Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers\u2019 equation. J. Sci. Comput. 53, 102\u2013128 (2012)","journal-title":"J. Sci. Comput."},{"key":"2839_CR16","doi-asserted-by":"publisher","first-page":"126","DOI":"10.1137\/110834901","volume":"50","author":"S Gottlieb","year":"2012","unstructured":"Gottlieb, S., Tone, F., Wang, C., Wang, X., Wirosoetisno, D.: Long time stability of a classical efficient scheme for two-dimensional Navier-Stokes equations. SIAM J. Numer. Anal. 50, 126\u2013150 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"2839_CR17","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1137\/040611434","volume":"43","author":"M Hochbruck","year":"2005","unstructured":"Hochbruck, 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":"2839_CR18","doi-asserted-by":"publisher","first-page":"4773","DOI":"10.1103\/PhysRevA.46.4773","volume":"46","author":"P Hohenberg","year":"1992","unstructured":"Hohenberg, P., Swift, J.: Effects of additive noise at the onset of Rayleigh-B\u00e9nard convection. Phys. Rev. A 46, 4773 (1992)","journal-title":"Phys. Rev. A"},{"key":"2839_CR19","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1016\/j.physd.2005.03.002","volume":"203","author":"A Hutt","year":"2005","unstructured":"Hutt, A., Atay, F.M.: Analysis of nonlocal neural fields for both general and gamma-distributed connectivities. Phys. D 203, 30\u201354 (2005)","journal-title":"Phys. D"},{"key":"2839_CR20","doi-asserted-by":"publisher","first-page":"755","DOI":"10.1016\/j.physd.2007.10.013","volume":"237","author":"A Hutt","year":"2008","unstructured":"Hutt, A., Longtin, A., Schimansky-Geier, L.: Additive noise-induced Turing transitions in spatial systems with application to neural fields and the SH equation. Phys. D 237, 755\u2013773 (2008)","journal-title":"Phys. D"},{"key":"2839_CR21","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1016\/j.cma.2018.08.019","volume":"343","author":"HG Lee","year":"2019","unstructured":"Lee, H.G.: An energy stable method for the SH equation with quadratic-cubic non-linearity. Comput. Methods Appl. Mech. Eng. 343, 40\u201351 (2019)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2839_CR22","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1016\/j.cma.2016.04.022","volume":"307","author":"HG Lee","year":"2016","unstructured":"Lee, H.G., Kim, J.: A simple and efficient finite difference method for the phase-field crystal equation on curved surfaces. Comput. Methods Appl. Mech. Eng. 307, 32\u201343 (2016)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2839_CR23","doi-asserted-by":"crossref","unstructured":"Li,X., Qiao,Z. Wang,C., Zheng,N.: Global-in-time energy stability for the exponential time differencing Runge\u2013Kutta scheme for the phase field crystal equation, arXiv preprint (2024), 2406. 06272","DOI":"10.1090\/mcom\/4067"},{"key":"2839_CR24","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-Hilliard equation. J. Sci. Comput. 70, 301\u2013341 (2017)","journal-title":"J. Sci. Comput."},{"key":"2839_CR25","doi-asserted-by":"publisher","first-page":"A429","DOI":"10.1137\/23M1552164","volume":"46","author":"X Li","year":"2024","unstructured":"Li, X., Qiao, Z.: A second-order, linear, $$L^\\infty $$-convergent, and energy stable scheme for the phase field crystal equation. SIAM J. Sci. Comput. 46, A429\u2013A451 (2024)","journal-title":"SIAM J. Sci. Comput."},{"key":"2839_CR26","first-page":"5","volume":"26","author":"X Li","year":"2019","unstructured":"Li, X., Ju, L., Meng, X.: Convergence analysis of exponential time differencing schemes for the Cahn-Hilliard equation. Commun. Comput. Phys. 26, 5 (2019)","journal-title":"Commun. Comput. Phys."},{"key":"2839_CR27","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-021-01519-7","volume":"87","author":"J Li","year":"2021","unstructured":"Li, J., Ju, L., Cai, Y., Feng, X.: Unconditionally maximum bound principle preserving linear schemes for the conservative Allen-Cahn equation with nonlocal constraint. J. Sci. Comput. 87, 1\u201332 (2021)","journal-title":"J. Sci. Comput."},{"key":"2839_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2022.114730","volume":"419","author":"Z Liu","year":"2023","unstructured":"Liu, Z., Chen, C.: On efficient semi-implicit auxiliary variable methods for the six-order SH model. J. Comput. Appl. Math. 419, 114730 (2023)","journal-title":"J. Comput. Appl. Math."},{"key":"2839_CR29","doi-asserted-by":"publisher","first-page":"957","DOI":"10.1090\/S0025-5718-08-02171-6","volume":"78","author":"S Maset","year":"2009","unstructured":"Maset, S., Zennaro, M.: Unconditionally stability of explicit exponential Runge-Kutta methods for semi-linear ordinary differential equations. Math. Comput. 78, 957\u2013967 (2009)","journal-title":"Math. Comput."},{"key":"2839_CR30","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1016\/j.apnum.2019.01.017","volume":"140","author":"S Pei","year":"2019","unstructured":"Pei, S., Hou, Y., You, B.: A linearly second-order energy stable scheme for the phase field crystal model. Appl. Numer. Math. 140, 134\u2013164 (2019)","journal-title":"Appl. Numer. Math."},{"key":"2839_CR31","doi-asserted-by":"publisher","first-page":"156","DOI":"10.1016\/S0378-4371(00)00144-8","volume":"283","author":"R Rosa","year":"2000","unstructured":"Rosa, R., Pont\u00e8s, J., Christov, C., Ramos, F.M., Neto, C.R., Rempel, E.L., Walgraef, D.: Gradient pattern analysis of SH dynamics: phase disorder characterization. Phys. A 283, 156\u2013159 (2000)","journal-title":"Phys. A"},{"key":"2839_CR32","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, J.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dynam. Systems 28, 1669\u20131691 (2010)","journal-title":"Discrete Contin. Dynam. Systems"},{"key":"2839_CR33","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112414","volume":"492","author":"J Sun","year":"2023","unstructured":"Sun, J., Zhang, H., Qian, X., Song, S.: A family of structure-preserving exponential time differencing Runge-Kutta schemes for the viscous Cahn-Hilliard equation. J. Comput. Phys. 492, 112414 (2023)","journal-title":"J. Comput. Phys."},{"key":"2839_CR34","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1103\/PhysRevA.15.319","volume":"15","author":"J Swift","year":"1977","unstructured":"Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319 (1977)","journal-title":"Phys. Rev. A"},{"key":"2839_CR35","doi-asserted-by":"publisher","first-page":"106","DOI":"10.4208\/csiam-am.SO-2024-0032","volume":"6","author":"H Wang","year":"2025","unstructured":"Wang, H., Sun, J., Zhang, H., Qian, X.: A Novel up to Fourth-Order Equilibria-Preserving and Energy-Stable Exponential Runge-Kutta Framework for Gradient Flows. CSIAM Trans. Appl. Math. 6, 106\u2013147 (2025)","journal-title":"CSIAM Trans. Appl. Math."},{"key":"2839_CR36","doi-asserted-by":"publisher","first-page":"2269","DOI":"10.1137\/080738143","volume":"47","author":"SM Wise","year":"2009","unstructured":"Wise, S.M., Wang, C., Lowengrub, J.S.: An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47, 2269\u20132288 (2009)","journal-title":"SIAM J. Numer. Anal."},{"key":"2839_CR37","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1007\/s10665-021-10122-6","volume":"128","author":"J Yang","year":"2021","unstructured":"Yang, J., Kim, J.: Linear and energy stable schemes for the SH equation with quadratic-cubic non-linearity based on a modified scalar auxiliary variable approach. J. Eng. Math. 128, 21 (2021)","journal-title":"J. Eng. Math."},{"key":"2839_CR38","doi-asserted-by":"publisher","first-page":"160","DOI":"10.1016\/j.camwa.2021.10.016","volume":"102","author":"J Yang","year":"2021","unstructured":"Yang, J., Tan, Z., Kim, J.: High-order time-accurate, efficient, and structure-preserving numerical methods for the conservative SH model. Comput. Math. Appl. 102, 160\u2013174 (2021)","journal-title":"Comput. Math. Appl."},{"key":"2839_CR39","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1051\/m2an\/2023101","volume":"58","author":"H Zhang","year":"2024","unstructured":"Zhang, H., Liu, L., Qian, X., Song, S.: Quantifying and eliminating the time delay in stabilization exponential time differencing Runge\u2013Kutta schemes for the Allen-Cahn equation. ESAIM Math. Modeling Numer. Anal. 58, 191\u2013221 (2024)","journal-title":"ESAIM Math. Modeling Numer. Anal."},{"key":"2839_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112708","volume":"499","author":"H Zhang","year":"2024","unstructured":"Zhang, H., Liu, L., Qian, X., Song, S.: Large time-stepping, delay-free, and invariant-set-preserving integrators for the viscous Cahn\u2013Hilliard\u2013Oono equation. J. Comput. Phys. 499, 112708 (2024)","journal-title":"J. Comput. Phys."},{"key":"2839_CR41","doi-asserted-by":"publisher","first-page":"2667","DOI":"10.1137\/24M1637623","volume":"62","author":"H Zhang","year":"2024","unstructured":"Zhang, H., Wang, H., Teng, X.: A second-order, global-in-time energy stable implicit-explicit Runge\u2013Kutta scheme for the phase field crystal equation. SIAM J. Numer. Anal. 62, 2667\u20132697 (2024)","journal-title":"SIAM J. Numer. Anal."},{"key":"2839_CR42","doi-asserted-by":"publisher","first-page":"1043","DOI":"10.1007\/s10915-015-0117-1","volume":"67","author":"L Zhu","year":"2016","unstructured":"Zhu, L., Ju, L., Zhao, W.: Fast high-order compact exponential time differencing Runge-Kutta methods for second-order semilinear parabolic equations. J. Sci. Comput. 67, 1043\u20131065 (2016)","journal-title":"J. Sci. Comput."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02839-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-025-02839-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02839-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,23]],"date-time":"2025-05-23T13:19:48Z","timestamp":1748006388000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-025-02839-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,16]]},"references-count":42,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["2839"],"URL":"https:\/\/doi.org\/10.1007\/s10915-025-02839-8","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,16]]},"assertion":[{"value":"24 September 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 February 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 April 2025","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 declare that they have no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"80"}}