{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T16:57:59Z","timestamp":1771001879999,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,11,2]],"date-time":"2023-11-02T00:00:00Z","timestamp":1698883200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071020"],"award-info":[{"award-number":["12071020"]}],"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":["12131005"],"award-info":[{"award-number":["12131005"]}],"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":["U2230402"],"award-info":[{"award-number":["U2230402"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>An efficient modified leapfrog time-marching scheme for the diblock copolymer model is investigated in this paper. The proposed scheme offers three main advantages. Firstly, it is linear in time, requiring only a linear algebra system to be solved at each time-marching step. This leads to a significant reduction in computational cost compared to other methods. Secondly, the scheme ensures unconditional energy stability, allowing for a large time step to be used without compromising solution stability. Thirdly, the existence and uniqueness of the numerical solution at each time step is rigorously proven, ensuring the reliability and accuracy of the method. A numerical example is also included to demonstrate and validate the proposed algorithm, showing its accuracy and efficiency in practical applications.<\/jats:p>","DOI":"10.3390\/computation11110215","type":"journal-article","created":{"date-parts":[[2023,11,2]],"date-time":"2023-11-02T09:16:21Z","timestamp":1698916581000},"page":"215","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Numerical Approximations of Diblock Copolymer Model Using a Modified Leapfrog Time-Marching Scheme"],"prefix":"10.3390","volume":"11","author":[{"given":"Lizhen","family":"Chen","sequence":"first","affiliation":[{"name":"Beijing Computational Science Research Center, Beijing 100193, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ying","family":"Ma","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Beijing University of Technology, Beijing 100124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bo","family":"Ren","sequence":"additional","affiliation":[{"name":"Beijing Computational Science Research Center, Beijing 100193, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guohui","family":"Zhang","sequence":"additional","affiliation":[{"name":"Beijing Computational Science Research Center, Beijing 100193, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1039\/B612304J","article-title":"Large scale alignment of a lamellar block copolymer thin film viaelectric fields: A time-resolved sfm study","volume":"2","author":"Olszowka","year":"2006","journal-title":"Soft Matter"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"031807","DOI":"10.1103\/PhysRevE.77.031807","article-title":"Phase-field modeling of the formation of lamellar nanostructures in diblock copolymer thin films under inplanar electric fields","volume":"77","author":"Wu","year":"2008","journal-title":"Phys. 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