{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T19:39:59Z","timestamp":1740166799867,"version":"3.37.3"},"reference-count":23,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100002261","name":"Russian Foundation for Basic Research","doi-asserted-by":"publisher","award":["14-01-00785"],"award-info":[{"award-number":["14-01-00785"]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Schemes with the second-order approximation in time are considered for numerically solving the Cauchy problem\nfor an evolutionary equation of first order with a self-adjoint operator.\nThe implicit two-level scheme based on the Pad\u00e9 polynomial approximation\nis unconditionally stable. It demonstrates good asymptotic properties in time\nand provides an adequate evolution in time for individual harmonics of the solution (has spectral mimetic (SM) stability).\nIn fact, the only drawback of this scheme is the necessity to solve\nan equation with an operator polynomial of second degree at each time level.\nWe consider modifications of these schemes, which are based on solving equations with operator polynomials of first degree.\nSuch computational implementations occur, for example,\nif we apply the fully implicit two-level scheme (the backward Euler scheme).\nA three-level modification of the SM-stable scheme is proposed. Its unconditional stability\nis established in the corresponding norms. The emphasis is on the scheme, where the numerical algorithm\ninvolves two stages, namely, the backward Euler scheme of first order at the first (prediction) stage\nand the following correction of the approximate solution using a factorized operator.\nThe SM-stability is established for the proposed scheme. To illustrate the theoretical results of the work,\na model problem is solved numerically.<\/jats:p>","DOI":"10.1515\/cmam-2016-0038","type":"journal-article","created":{"date-parts":[[2016,11,22]],"date-time":"2016-11-22T10:01:02Z","timestamp":1479808862000},"page":"323-335","source":"Crossref","is-referenced-by-count":0,"title":["Factorized Schemes of Second-Order Accuracy for Numerically Solving Unsteady Problems"],"prefix":"10.1515","volume":"17","author":[{"given":"Petr N.","family":"Vabishchevich","sequence":"first","affiliation":[{"name":"Nuclear Safety Institute, Russian Academy of Sciences, 52 B. Tulskaya, 115191 Moscow; and North-Eastern Federal University, 58 Belinskogo, 677000 Yakutsk, Russia"}]}],"member":"374","published-online":{"date-parts":[[2016,11,22]]},"reference":[{"key":"2023033114222322510_j_cmam-2016-0038_ref_001_w2aab3b7d360b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"Ascher U. M.,\nNumerical Methods for Evolutionary Differential Equations,\nSociety for Industrial Mathematics, Philadelphia, 2008.","DOI":"10.1137\/1.9780898718911"},{"key":"2023033114222322510_j_cmam-2016-0038_ref_002_w2aab3b7d360b1b6b1ab2ab2Aa","unstructured":"Baker G. A. and Graves-Morris P. R.,\nPad\u00e9 Approximants,\nCambridge University Press, Cambridge, 1996."},{"key":"2023033114222322510_j_cmam-2016-0038_ref_003_w2aab3b7d360b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"Butcher J. 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W.,\nNumerical Initial Value Problems in Ordinary Differential Equations,\nPrentice Hall, Englewood Cliffs, 1971."},{"key":"2023033114222322510_j_cmam-2016-0038_ref_006_w2aab3b7d360b1b6b1ab2ab6Aa","unstructured":"Gustafsson B.,\nHigh Order Difference Methods for Time Dependent PDE,\nSpringer, Berlin, 2008."},{"key":"2023033114222322510_j_cmam-2016-0038_ref_007_w2aab3b7d360b1b6b1ab2ab7Aa","unstructured":"Hairer E. and Wanner G.,\nSolving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems,\nSpringer, Berlin, 2010."},{"key":"2023033114222322510_j_cmam-2016-0038_ref_008_w2aab3b7d360b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"Higham N. J.,\nFunctions of Matrices: Theory and Computation,\nSIAM, Philadelphia, 2008.","DOI":"10.1137\/1.9780898717778"},{"key":"2023033114222322510_j_cmam-2016-0038_ref_009_w2aab3b7d360b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"Hundsdorfer W. H. and Verwer J. 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G.,\nNumerical Methods for Solving Stiff Systems (in Russian),\nNauka, Moscow, 1979."},{"key":"2023033114222322510_j_cmam-2016-0038_ref_013_w2aab3b7d360b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"Samarskii A. A.,\nThe Theory of Difference Schemes,\nMarcel Dekker, New York, 2001.","DOI":"10.1201\/9780203908518"},{"key":"2023033114222322510_j_cmam-2016-0038_ref_014_w2aab3b7d360b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"Samarskii A. A., Gavrilyuk I. P. and Makarov V. L.,\nStability and regularization of three-level difference schemes with unbounded operator coefficients in Banach spaces,\nSIAM J. Numer. Anal. 38 (2001), no. 2, 708\u2013723.","DOI":"10.1137\/S0036142999357221"},{"key":"2023033114222322510_j_cmam-2016-0038_ref_015_w2aab3b7d360b1b6b1ab2ac15Aa","unstructured":"Samarskii A. A. and Gulin A. V.,\nStability of Difference Schemes (in Russian),\nNauka, Moscow, 1973."},{"key":"2023033114222322510_j_cmam-2016-0038_ref_016_w2aab3b7d360b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"Samarskii A. A., Matus P. P. and Vabishchevich P. N.,\nDifference Schemes with Operator Factors,\nKluwer Academic, Dordrecht, 2002.","DOI":"10.1007\/978-94-015-9874-3"},{"key":"2023033114222322510_j_cmam-2016-0038_ref_017_w2aab3b7d360b1b6b1ab2ac17Aa","unstructured":"Samarskii A. A. and Vabishchevich P. N.,\nComputational Heat Transfer,\nWiley, Chichester, 1995."},{"key":"2023033114222322510_j_cmam-2016-0038_ref_018_w2aab3b7d360b1b6b1ab2ac18Aa","doi-asserted-by":"crossref","unstructured":"Vabishchevich P. N.,\nFactorized SM-stable two-level schemes,\nComput. Math. Math. Phys. 50 (2010), no. 11, 1818\u20131824.","DOI":"10.1134\/S0965542510110059"},{"key":"2023033114222322510_j_cmam-2016-0038_ref_019_w2aab3b7d360b1b6b1ab2ac19Aa","doi-asserted-by":"crossref","unstructured":"Vabishchevich P. 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