{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:57:07Z","timestamp":1760230627493,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,2]],"date-time":"2022-08-02T00:00:00Z","timestamp":1659398400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Fundamental Research Funds for the Central Universities of Central South University","award":["1053320192590"],"award-info":[{"award-number":["1053320192590"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Poisson\u2013Nernst\u2013Planck (PNP) system is a nonlinear coupled system that describes the motion of ionic particles. As the exact solution of the system is not available, numerical investigations are essentially important, and there are quite a lot of numerical methods proposed in the existing literature. However, the theoretical analysis is usually neglected due to the complicated nature of the PNP system. In this paper, a theoretical investigation for a symmetrical finite difference method proposed in the previous literature was conducted. An L2 error estimate of O(\u03c4+h2) was derived for the numerical scheme in 1D, where \u03c4 denotes the time step size and h denotes the spatial mesh size, respectively. Numerical results confirm the theoretical analysis. More importantly, a positivity-preserving condition for the scheme is provided with rigorously theoretical justification.<\/jats:p>","DOI":"10.3390\/sym14081589","type":"journal-article","created":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T00:15:26Z","timestamp":1659485726000},"page":"1589","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Convergence Analysis of a Symmetrical and Positivity-Preserving Finite Difference Scheme for 1D Poisson\u2013Nernst\u2013Planck System"],"prefix":"10.3390","volume":"14","author":[{"given":"Weiwei","family":"Ling","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China"},{"name":"School of Social Management, Jiangxi College of Applied Technology, Ganzhou 341000, China"}]},{"given":"Benchao","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Qilu Transportation, Shandong University, Jinan 250002, China"}]},{"given":"Qian","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Water Conservancy and Environment, University of Jinan, Jinan 250022, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1137\/060667335","article-title":"Imcompressible ionized non-newtonian fluid mixture","volume":"39","author":"Roubick","year":"2007","journal-title":"SIAM J. 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[2nd ed.]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1589\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:01:15Z","timestamp":1760140875000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1589"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,2]]},"references-count":13,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081589"],"URL":"https:\/\/doi.org\/10.3390\/sym14081589","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,2]]}}}