{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T00:50:23Z","timestamp":1761094223026,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T00:00:00Z","timestamp":1760918400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["24-41-02035"],"award-info":[{"award-number":["24-41-02035"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>The DuFort\u2013Frankel scheme was introduced in the 1950s to solve parabolic equations, and has been widely used ever since due to its stability and explicit nature. However, for over seven decades, its application has been limited to Cartesian grids. In this work, we propose a generalization of the DuFort\u2013Frankel scheme that could be applied to arbitrary unstructured grids. Specifically, we focus on Voronoi grids in both 2D and 3D, and use the finite volume method for spatial discretization. Additionally, we present a proof of its stability based on the analysis of the spectrum of the amplification matrix, along with numerical examples.<\/jats:p>","DOI":"10.3390\/computation13100246","type":"journal-article","created":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T13:54:41Z","timestamp":1760968481000},"page":"246","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Stability of the DuFort\u2013Frankel Scheme on Unstructured Grids"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8913-7710","authenticated-orcid":false,"given":"Nikolay","family":"Yavich","sequence":"first","affiliation":[{"name":"AI Center, Skolkovo Institute of Science and Technology, Moscow 121205, Russia"}]},{"given":"Evgeny","family":"Burnaev","sequence":"additional","affiliation":[{"name":"AI Center, Skolkovo Institute of Science and Technology, Moscow 121205, Russia"},{"name":"Autonomous Non-Profit Organization Artificial Intelligence Research Institute (AIRI), Moscow 105064, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3317-9572","authenticated-orcid":false,"given":"Vladimir","family":"Vanovskiy","sequence":"additional","affiliation":[{"name":"AI Center, Skolkovo Institute of Science and Technology, Moscow 121205, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"135","DOI":"10.2307\/2002754","article-title":"Stability Conditions in the Numerical Treatment of Parabolic Differential Equations","volume":"7","author":"Frankel","year":"1953","journal-title":"Math. 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