{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T06:47:01Z","timestamp":1774594021281,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T00:00:00Z","timestamp":1663804800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science Committee of the Ministry of Education","award":["AP08856594"],"award-info":[{"award-number":["AP08856594"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The paper is devoted to the construction and study of a numerical method for solving two-dimensional Saint\u2013Venant equations. These equations have important applied significance in modern hydraulic engineering and are suitable for describing waves in the atmosphere, rivers and oceans, and for modeling tides. The issues of formulation of the mixed problems for these equations are studied. The system of equations is reduced to a symmetrical form by transforming dependent variables. Then, the matrices of coefficients are represented as the sums of two symmetric semidefinite matrices. This transformation allows constructing an upwind difference scheme in spatial directions to determine the numerical solution of the initial boundary value problem. The stability of the proposed difference scheme in energy norms is rigorously proved. The results of numerical experiments conducted for a model problem are provided to confirm the stability of the proposed method.<\/jats:p>","DOI":"10.3390\/sym14101986","type":"journal-article","created":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T04:07:07Z","timestamp":1663906027000},"page":"1986","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Stability Analysis of an Upwind Difference Splitting Scheme for Two-Dimensional Saint\u2013Venant Equations"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1228-8246","authenticated-orcid":false,"given":"Abdumauvlen","family":"Berdyshev","sequence":"first","affiliation":[{"name":"Department of Mathematics and Mathematical Modelling, Institute of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, Almaty 050000, Kazakhstan"},{"name":"Institute of Information and Computational Technologies SC MES, Almaty 050010, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4734-3070","authenticated-orcid":false,"given":"Rakhmatillo","family":"Aloev","sequence":"additional","affiliation":[{"name":"Department of Computational Mathematics and Information Systems, Faculty of Applied Mathematics and Intellectual Technology, Ulugbek National University of Uzbekistan, Tashkent 100174, Uzbekistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7703-0329","authenticated-orcid":false,"given":"Dana","family":"Bliyeva","sequence":"additional","affiliation":[{"name":"Institute of Information and Computational Technologies SC MES, Almaty 050010, Kazakhstan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7296-953X","authenticated-orcid":false,"given":"Sardor","family":"Dadabayev","sequence":"additional","affiliation":[{"name":"Department of Information Technologies, Faculty of Information Technologies and Computer Engineering, Andijan State University, Andijan 170100, Uzbekistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4812-4104","authenticated-orcid":false,"given":"Zharasbek","family":"Baishemirov","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Mathematical Modelling, Institute of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, Almaty 050000, Kazakhstan"},{"name":"Institute of Information and Computational Technologies SC MES, Almaty 050010, Kazakhstan"},{"name":"School of Applied Mathematics, Kazakh-British Technical University, Almaty 050000, Kazakhstan"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1193","DOI":"10.4236\/jamp.2020.87090","article-title":"Study of the 1D Saint-Venant Equations and Application to the Simulation of a Flood Problem","volume":"8","author":"Kader","year":"2020","journal-title":"J. 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