{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T22:22:06Z","timestamp":1770243726478,"version":"3.49.0"},"reference-count":34,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T00:00:00Z","timestamp":1770163200000},"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":["11571053"],"award-info":[{"award-number":["11571053"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Postgraduate Scientific Research Innovation Project of Hunan Province","award":["CX20230923"],"award-info":[{"award-number":["CX20230923"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This study proposes an efficient numerical scheme for simulating heat transfer governed by the diffusion equation with moving singular sources. The work addresses two-dimensional problems with line sources and three-dimensional problems with plane sources, which are prevalent in irreversible thermodynamic processes. Developed within a finite difference framework, the method employs a partitioned discretization strategy to accurately resolve the solution singularity near the heat source\u2014a region critical for precise local entropy production analysis. In the immediate vicinity of the source, we analytically derive and incorporate the solution\u2019s \u201cjump\u201d conditions to construct specialized finite difference approximations. Away from the source, standard second-order-accurate schemes are applied. This hybrid approach yields a globally second-order convergent spatial discretization. The resulting sparse system is efficient for large-scale simulation of dissipative systems. The accuracy and efficacy of the proposed method are demonstrated through numerical examples, providing a reliable tool for the detailed study of energy distribution in non-equilibrium thermal processes.<\/jats:p>","DOI":"10.3390\/e28020179","type":"journal-article","created":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T09:02:01Z","timestamp":1770195721000},"page":"179","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Partitioned Finite Difference Method for Heat Transfer with Moving Line and Plane Heat Sources"],"prefix":"10.3390","volume":"28","author":[{"given":"Jun","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yingjun","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.optlastec.2004.02.017","article-title":"Thermal modelling of laser welding and related processes: A literature review","volume":"37","author":"Mackwood","year":"2005","journal-title":"Opt. 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