{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T03:14:27Z","timestamp":1768533267003,"version":"3.49.0"},"reference-count":6,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,10,1]]},"abstract":"Abstract<\/jats:title>\n The convergence of difference schemes on uniform grids for an initial-boundary value problem for a singularly perturbed parabolic convection-diffusion equation is studied; the highest x<\/jats:italic>-derivative in the equation is multiplied by a perturbation parameter \u03b5 taking arbitrary values in the interval \n \n \n \n (<\/m:mo>\n 0<\/m:mn>\n ,<\/m:mo>\n 1<\/m:mn>\n ]<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {(0,1]}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nFor small \u03b5, the problem involves a boundary layer of width \n \n \n \n \ud835\udcaa<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03b5<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{O}(\\varepsilon)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where the solution changes by a finite value, while its derivative grows unboundedly as \u03b5 tends to zero.\nWe construct a standard difference scheme on uniform meshes based on the classical monotone grid approximation (upwind approximation of the first-order derivatives).\nUsing a priori estimates, we show that such a scheme converges as \n \n \n \n \n \n {<\/m:mo>\n \n \u03b5<\/m:mi>\n \u2062<\/m:mo>\n N<\/m:mi>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n ,<\/m:mo>\n \n N<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n <\/m:mrow>\n \u2192<\/m:mo>\n \u221e<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n {\\{\\varepsilon N\\},N_{0}\\to\\infty}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in the maximum norm with first-order accuracy in \n \n \n \n {<\/m:mo>\n \n \u03b5<\/m:mi>\n \u2062<\/m:mo>\n N<\/m:mi>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {\\{\\varepsilon N\\}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n N<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n {N_{0}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>; as \n \n \n \n \n N<\/m:mi>\n ,<\/m:mo>\n \n N<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n <\/m:mrow>\n \u2192<\/m:mo>\n \u221e<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n {N,N_{0}\\to\\infty}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, the convergence is conditional with respect to<\/jats:italic>\n N<\/jats:italic>, where \n \n \n \n N<\/m:mi>\n +<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {N+1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n \n N<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n +<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {N_{0}+1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are the numbers of mesh points in x<\/jats:italic> and t<\/jats:italic>, respectively.\nWe develop an improved<\/jats:italic> difference scheme on uniform meshes using the grid approximation of the first x<\/jats:italic>-derivative in the convective term by the central<\/jats:italic> difference operator under the condition<\/jats:italic>\n \n \n \n \n h<\/m:mi>\n \u2264<\/m:mo>\n \n m<\/m:mi>\n \u2062<\/m:mo>\n \u03b5<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {h\\leq m\\varepsilon}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, which ensures the monotonicity of the scheme; here m<\/jats:italic> is some rather small positive constant.\nIt is proved that this scheme converges in the maximum norm at a rate<\/jats:italic> of \n \n \n \n \ud835\udcaa<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \n \n \u03b5<\/m:mi>\n \n -<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n \u2062<\/m:mo>\n \n N<\/m:mi>\n \n -<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n <\/m:mrow>\n +<\/m:mo>\n \n N<\/m:mi>\n 0<\/m:mn>\n \n -<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:msubsup>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{O}(\\varepsilon^{-2}N^{-2}+N^{-1}_{0})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nWe compare the convergence rate of the developed scheme with the known Samarskii scheme for a regular<\/jats:italic> problem.\nIt is found that the improved scheme (for \n \n \n \n \u03b5<\/m:mi>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {\\varepsilon=1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>), as well as the Samarskii scheme, converges in the maximum norm<\/jats:italic> with second-order accuracy in<\/jats:italic>\n x<\/jats:italic> and first-order accuracy in<\/jats:italic>\n t<\/jats:italic>.<\/jats:p>","DOI":"10.1515\/cmam-2019-0023","type":"journal-article","created":{"date-parts":[[2019,6,14]],"date-time":"2019-06-14T09:10:56Z","timestamp":1560503456000},"page":"709-715","source":"Crossref","is-referenced-by-count":6,"title":["Difference Schemes on Uniform Grids for an Initial-Boundary Value Problem for a Singularly Perturbed Parabolic Convection-Diffusion Equation"],"prefix":"10.1515","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6886-8979","authenticated-orcid":false,"given":"Grigorii I.","family":"Shishkin","sequence":"first","affiliation":[{"name":"Krasovskii Institute of Mathematics and Mechanics , Ural Branch, Russian Academy of Sciences, 620990 Yekaterinburg , Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8489-5432","authenticated-orcid":false,"given":"Lidia P.","family":"Shishkina","sequence":"additional","affiliation":[{"name":"Krasovskii Institute of Mathematics and Mechanics , Ural Branch, Russian Academy of Sciences, 620990 Yekaterinburg , Russia"}]}],"member":"374","published-online":{"date-parts":[[2019,6,13]]},"reference":[{"key":"2023033110450842779_j_cmam-2019-0023_ref_001","doi-asserted-by":"crossref","unstructured":"P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O\u2019Riordan and G. I. Shishkin,\nRobust Computational Techniques for Boundary Layers,\nChapman & Hall\/CRC Press, Boca Raton, 2000.","DOI":"10.1201\/9781482285727"},{"key":"2023033110450842779_j_cmam-2019-0023_ref_002","doi-asserted-by":"crossref","unstructured":"J. J. H. Miller, E. O\u2019Riordan and G. I. Shishkin,\nFitted Numerical Methods for Singular Perturbation Problems. Error Estimates in Maximum Norm for Linear Problems in One and Two Dimentions,\nWorld Scientific, Singapore, 2012.","DOI":"10.1142\/8410"},{"key":"2023033110450842779_j_cmam-2019-0023_ref_003","doi-asserted-by":"crossref","unstructured":"A. A. Samarskii,\nMonotonic difference schemes for elliptic and parabolic equations in the case of a non-selfadjoint elliptic operator,\nUSSR Comput. Math. Math. Phys. 5 (1965), no. 3, 212\u2013217.","DOI":"10.1016\/0041-5553(65)90158-8"},{"key":"2023033110450842779_j_cmam-2019-0023_ref_004","unstructured":"A. A. Samarskii,\nThe Theory of Difference Schemes (in Russian), 3rd ed.,\nNauka, Moscow, 1989."},{"key":"2023033110450842779_j_cmam-2019-0023_ref_005","unstructured":"G. I. Shishkin,\nDiscrete Approximations of Singularly Perturbed Elliptic and Parabolic Equations (in Russian),\nRussian Academy of Sciences, Ekaterinburg, 1992."},{"key":"2023033110450842779_j_cmam-2019-0023_ref_006","doi-asserted-by":"crossref","unstructured":"G. I. Shishkin and L. P. Shishkina,\nDifference Methods for Singular Perturbation Problems,\nCRC Press, Boca Raton, 2009.","DOI":"10.1201\/9780203492413"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/20\/4\/article-p709.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0023\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0023\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T12:57:10Z","timestamp":1680267430000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0023\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,13]]},"references-count":6,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,8,5]]},"published-print":{"date-parts":[[2020,10,1]]}},"alternative-id":["10.1515\/cmam-2019-0023"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2019-0023","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,13]]}}}