{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T23:11:18Z","timestamp":1771024278568,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T00:00:00Z","timestamp":1763596800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"NSF","doi-asserted-by":"publisher","award":["2424826\/2424827"],"award-info":[{"award-number":["2424826\/2424827"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We introduce a hybridizable discontinuous Galerkin (HDG) scheme for solving the Poisson\u2013Nernst\u2013Planck (PNP) equations. The log-density formulation as introduced by Metti et al. in their paper \u201cEnergetically stable discretizations for charge transport and electrokinetic models. J. Comput. Phys. 2016, 306, 1-18\u201d is utilized to ensure the positivity of the densities of the charged particles. We further prove that our fully discrete scheme is energy stable and mass conserving. Numerical simulations are provided to demonstrate the accuracy of the scheme in one and two spatial dimensions. A derivation of an HDG-DG space\u2013time scheme is given, with implementation and convergence analysis left to future work.<\/jats:p>","DOI":"10.3390\/e27111175","type":"journal-article","created":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T12:02:42Z","timestamp":1763640162000},"page":"1175","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Positivity-Preserving Hybridizable Discontinuous Galerkin Scheme for Solving PNP Model"],"prefix":"10.3390","volume":"27","author":[{"given":"Diana","family":"Morales","sequence":"first","affiliation":[{"name":"Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4056-6933","authenticated-orcid":false,"given":"Zhiliang","family":"Xu","sequence":"additional","affiliation":[{"name":"Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"779","DOI":"10.4310\/CMS.2014.v12.n4.a9","article-title":"An Energetic Variational Approach for ion transport","volume":"12","author":"Xu","year":"2014","journal-title":"Commun. 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