{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:58:32Z","timestamp":1760144312004,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,9]],"date-time":"2024-04-09T00:00:00Z","timestamp":1712620800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The aim of this article is to analyze the efficiency and accuracy of finite-difference and finite-element Galerkin schemes for non-stationary hyperbolic and parabolic problems. The main problem solved in this article deals with the construction of accurate and efficient discrete schemes on nonuniform and dynamic grids in time and space. The presented stability and convergence analysis enables improving the existing accuracy estimates. The obtained stability results show explicitly the rate of accumulation of interpolation and projection errors that arise due to the movement of grid points. It is shown that the cases when the time grid steps are doubled or halved have different stability properties. As an additional technique to improve the accuracy of discretizations on non-stationary space grids, it is recommended to use projection operators instead of interpolation operators. This technique is used to solve a test parabolic problem. The results of specially selected computational experiments are also presented, and they confirm the accuracy of all theoretical error estimates obtained in this article.<\/jats:p>","DOI":"10.3390\/axioms13040244","type":"journal-article","created":{"date-parts":[[2024,4,9]],"date-time":"2024-04-09T04:37:27Z","timestamp":1712637447000},"page":"244","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On a Stability of Non-Stationary Discrete Schemes with Respect to Interpolation Errors"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3262-3048","authenticated-orcid":false,"given":"Raimondas","family":"\u010ciegis","sequence":"first","affiliation":[{"name":"Mathematical Modelling Department, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saul\u0117tekio al. 11, LT-10223 Vilnius, Lithuania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Olga","family":"Subo\u010d","sequence":"additional","affiliation":[{"name":"Mathematical Modelling Department, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saul\u0117tekio al. 11, LT-10223 Vilnius, Lithuania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Remigijus","family":"\u010ciegis","sequence":"additional","affiliation":[{"name":"Kaunas Faculty, Vilnius University, Muitin\u0117s St 8, LT-44280 Kaunas, Lithuania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hundsdorfer, W., and Verwer, J. 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