{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:39:55Z","timestamp":1760060395102,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T00:00:00Z","timestamp":1755734400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key Scientific Research Projects of Universities in Henan Province","award":["26A110004","23YJCGJW008","242400411163","2025-YYZD-10"],"award-info":[{"award-number":["26A110004","23YJCGJW008","242400411163","2025-YYZD-10"]}]},{"name":"Humanities and Social Science Foundation of the Ministry of Education of China","award":["26A110004","23YJCGJW008","242400411163","2025-YYZD-10"],"award-info":[{"award-number":["26A110004","23YJCGJW008","242400411163","2025-YYZD-10"]}]},{"name":"Henan Provincial Science and Technology Department Project","award":["26A110004","23YJCGJW008","242400411163","2025-YYZD-10"],"award-info":[{"award-number":["26A110004","23YJCGJW008","242400411163","2025-YYZD-10"]}]},{"name":"Major Project of Philosophy and Social Research in Colleges and Universities of Henan Province","award":["26A110004","23YJCGJW008","242400411163","2025-YYZD-10"],"award-info":[{"award-number":["26A110004","23YJCGJW008","242400411163","2025-YYZD-10"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, high-order compact difference methods (HOCDMs) are proposed to solve the semi-linear Sobolev equations (SLSEs), which arise in various physical models, such as porous media flow and heat conduction. First, a two-level numerical method is given by applying the Crank\u2013Nicolson (C-N) method in time and the fourth-order compact difference method in space. This method is shown to achieve second-order accuracy in time and fourth-order accuracy in space. Subsequently, we introduce the Richardson extrapolation technique to improve the temporal accuracy of the two-level method from second order to fourth order. Furthermore, we devise a fully fourth-order method in both time and space by applying the fourth-order difference method to discretize both temporal and spatial derivatives, and we provide a proof of its convergence. Finally, a series of numerical experiments is conducted to verify the effectiveness of the proposed methods.<\/jats:p>","DOI":"10.3390\/axioms14080646","type":"journal-article","created":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T15:19:02Z","timestamp":1755789542000},"page":"646","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["High-Order Time\u2013Space Compact Difference Methods for Semi-Linear Sobolev Equations"],"prefix":"10.3390","volume":"14","author":[{"given":"Bo","family":"Hou","sequence":"first","affiliation":[{"name":"School of Business, Henan University of Science and Technology, Luoyang 471000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tianhua","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Business, Henan University of Science and Technology, Luoyang 471000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9613-8386","authenticated-orcid":false,"given":"Guoqu","family":"Deng","sequence":"additional","affiliation":[{"name":"School of Business, Henan University of Science and Technology, Luoyang 471000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhi","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Sichuan Agricultural University, Ya\u2019an 625000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1286","DOI":"10.1016\/0021-8928(60)90107-6","article-title":"Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks","volume":"24","author":"Barenblatt","year":"1960","journal-title":"J. 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