{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T23:26:21Z","timestamp":1768519581343,"version":"3.49.0"},"reference-count":34,"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 For Dirichlet initial boundary value problem (IBVP) for two-dimensional\nquasilinear parabolic equations with mixed derivatives monotone linearized difference scheme is constructed.\nThe ellipticity conditions<\/jats:p>\n \n \n \n \n \n \n \n c<\/m:mi>\n 1<\/m:mn>\n <\/m:msub>\n \u2062<\/m:mo>\n \n \n \u2211<\/m:mo>\n \n \u03b1<\/m:mi>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n 2<\/m:mn>\n <\/m:munderover>\n \n \u03be<\/m:mi>\n \u03b1<\/m:mi>\n 2<\/m:mn>\n <\/m:msubsup>\n <\/m:mrow>\n <\/m:mrow>\n \u2264<\/m:mo>\n \n \n \u2211<\/m:mo>\n \n \n \u03b1<\/m:mi>\n ,<\/m:mo>\n \u03b2<\/m:mi>\n <\/m:mrow>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n 2<\/m:mn>\n <\/m:munderover>\n \n \n k<\/m:mi>\n \n \u03b1<\/m:mi>\n \u2062<\/m:mo>\n \u03b2<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n u<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n \u03be<\/m:mi>\n \u03b1<\/m:mi>\n <\/m:msub>\n \u2062<\/m:mo>\n \n \u03be<\/m:mi>\n \u03b2<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n <\/m:mrow>\n \u2264<\/m:mo>\n \n \n c<\/m:mi>\n 2<\/m:mn>\n <\/m:msub>\n \u2062<\/m:mo>\n \n \n \u2211<\/m:mo>\n \n \u03b1<\/m:mi>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n 2<\/m:mn>\n <\/m:munderover>\n \n \u03be<\/m:mi>\n \u03b1<\/m:mi>\n 2<\/m:mn>\n <\/m:msubsup>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n c_{1}\\sum_{\\alpha=1}^{2}\\xi_{\\alpha}^{2}\\leq\\sum_{\\alpha,\\beta=1}^{2}k_{\\alpha%\n\\beta}(u)\\xi_{\\alpha}\\xi_{\\beta}\\leq c_{2}\\sum_{\\alpha=1}^{2}\\xi_{\\alpha}^{2}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:disp-formula>\n <\/jats:p>\n are assumed to be fulfilled for the sign alternating solution \n \n \n \n \n u<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \ud835\udc31<\/m:mi>\n ,<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \u2208<\/m:mo>\n \n \n D<\/m:mi>\n \u00af<\/m:mo>\n <\/m:mover>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n u<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {u(\\mathbf{x},t)\\in\\bar{D}(u)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> only in the domain of exact solution values (unbounded nonlinearity).\nOn the basis of the proved new corollaries of the maximum principle, not only two-sided estimates for the approximate solution y<\/jats:italic> but also its belonging to the domain of exact solution values are established.\nWe assume that the solution is continuous and its first derivatives \n \n \n \n \n \u2202<\/m:mo>\n \u2061<\/m:mo>\n u<\/m:mi>\n <\/m:mrow>\n \n \u2202<\/m:mo>\n \u2061<\/m:mo>\n \n x<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n <\/m:mfrac>\n <\/m:math>\n \n {\\frac{\\partial u}{\\partial x_{i}}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> have discontinuities of the first kind in the neighborhood of the finite number of discontinuity lines.\nNo smoothness of the time derivative is assumed.\nThe convergence of an approximate solution to a generalized solution of a differential problem in the grid norm \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n {L_{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is proved.<\/jats:p>","DOI":"10.1515\/cmam-2019-0052","type":"journal-article","created":{"date-parts":[[2019,7,19]],"date-time":"2019-07-19T09:03:22Z","timestamp":1563527002000},"page":"695-707","source":"Crossref","is-referenced-by-count":3,"title":["On Convergence of Difference Schemes for Dirichlet IBVP for Two-Dimensional Quasilinear Parabolic Equations with Mixed Derivatives and Generalized Solutions"],"prefix":"10.1515","volume":"20","author":[{"given":"Piotr","family":"Matus","sequence":"first","affiliation":[{"name":"Institute of Mathematics and Computer Science , The John Paul II Catholic University of Lublin , Al. Raclawickie 14, 20-950 Lublin , Poland ; and Institute of Mathematics, NAS of Belarus, 11 Surganov St., 220072 Minsk, Belarus"}]},{"given":"Dmitriy","family":"Poliakov","sequence":"additional","affiliation":[{"name":"Institute of Mathematics , NAS of Belarus , 11 Surganov St., 220072 Minsk , Belarus"}]},{"given":"Le Minh","family":"Hieu","sequence":"additional","affiliation":[{"name":"University of Economics , The University of Danang , 71 Ngu Hanh Son Str., 550000 Danang , Vietnam"}]}],"member":"374","published-online":{"date-parts":[[2019,7,19]]},"reference":[{"key":"2023033110450840980_j_cmam-2019-0052_ref_001","unstructured":"V. N. Abra\u0161in,\nDifference schemes for a nonlinear parabolic equation that is not solved for the highest derivatives,\nDiffer. Equ. 11 (1976), 524\u2013533."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_002","unstructured":"V. N. Abra\u0161in,\nDifference schemes for nonlinear hyperbolic equations. II,\nDiffer. Equ. 11 (1976), 224\u2013235."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_003","doi-asserted-by":"crossref","unstructured":"G. Akrivis, M. Crouzeix and C. Makridakis,\nImplicit-explicit multistep methods for quasilinear parabolic equations,\nNumer. Math. 82 (1999), no. 4, 521\u2013541.","DOI":"10.1007\/s002110050429"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_004","unstructured":"A. A. Amosov and A. A. Zlotnik,\nA difference scheme for equations of the one-dimensional movement of a viscous barotropic gas,\nComputational Processes and Systems. No. 4 (in Russian),\n\u201cNauka\u201d, Moscow (1986), 192\u2013218."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_005","doi-asserted-by":"crossref","unstructured":"A. A. Amosov and A. A. Zlotnik,\nDifference schemes of the second order of accuracy for equations of the one-dimensional motion of a viscous gas,\nUSSR Comput. Math. Math. Phys. 27 (1987), no. 4, 46\u201357.","DOI":"10.1016\/0041-5553(87)90008-5"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_006","doi-asserted-by":"crossref","unstructured":"I. Farag\u00f3 and R. Horv\u00e1th,\nDiscrete maximum principle and adequate discretizations of linear parabolic problems,\nSIAM J. Sci. Comput. 28 (2006), no. 6, 2313\u20132336.","DOI":"10.1137\/050627241"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_007","doi-asserted-by":"crossref","unstructured":"I. Farag\u00f3, J. Kar\u00e1tson and S. Korotov,\nDiscrete maximum principles for nonlinear parabolic PDE systems,\nIMA J. Numer. Anal. 32 (2012), no. 4, 1541\u20131573.","DOI":"10.1093\/imanum\/drr050"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_008","unstructured":"S. K. Godunov,\nA difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics (in Russian),\nMat. Sb. (N.S.) 47(89) (1959), 271\u2013306."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_009","unstructured":"B. S. \u012covanovich, P. P. Matus and V. S. Shcheglik,\nOn the accuracy of difference schemes on nonlinear parabolic equations with generalized solutions,\nComput. Math. Math. Phys. 39 (1999), no. 10, 1611\u20131618."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_010","unstructured":"M. M. Karchevsky and M. F. Pavlova,\nUravneniya matematicheskoy fiziki. Dopolnitelnye glavy,\nIzdatel\u2019stvo Kazanskogo Gosudarstvennogo Universiteta, Kazan, 2008."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_011","unstructured":"O. A. Lady\u017eenskaya,\nSolution of the first boundary problem in the large for quasi-linear parabolic equations (in Russian),\nTrudy Moskov. Mat. Ob\u0161\u010d. 7 (1958), 149\u2013177."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_012","unstructured":"O. A. Lady\u017eenskaya, V. A. Solonnikov and N. N. Ural\u2019ceva,\nLinear and Quasilinear Equations of Parabolic Type,\nTransl. Math. Monogr. 23,\nAmerican Mathematical Society, Providence, 1968."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_013","doi-asserted-by":"crossref","unstructured":"R. D. Lazarov, V. L. Makarov and W. Weinelt,\nOn the convergence of difference schemes for the approximation of solutions \n \n \n \n u\n \u2208\n \n W\n 2\n m\n \n \n \n \n u\\in W^{m}_{2}\n \n \n \n \n \n \n (\n \n m\n &\n \n g\n \u2062\n t\n \n \n ;\n 0.5\n )\n \n \n \n (m\\>0.5)\n \n of elliptic equations with mixed derivatives,\nNumer. Math. 44 (1984), no. 2, 223\u2013232.","DOI":"10.1007\/BF01410107"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_014","doi-asserted-by":"crossref","unstructured":"F. V. Lubyshev and M. E. Fairuzov,\nConsistent convergence rate estimates in the grid \n \n \n \n \n W\n \n 2\n ,\n 0\n \n 2\n \n \u2062\n \n (\n \u03c9\n )\n \n \n \n \n W_{2,0}^{2}(\\omega)\n \n norm for difference schemes approximating nonlinear elliptic equations with mixed derivatives and solutions from \n \n \n \n \n W\n \n 2\n ,\n 0\n \n m\n \n \u2062\n \n (\n \u03a9\n )\n \n \n \n \n W_{2,0}^{m}(\\Omega)\n \n , \n \n \n \n \n \n 3\n &\n \n l\n \u2062\n t\n \n \n ;\n m\n \n \u2264\n 4\n \n \n \n 3\\<m\\leq 4\n \n ,\nComput. Math. Math. Phys. 57 (2017), no. 9, 1427\u20131452.","DOI":"10.1134\/S0965542517090081"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_015","doi-asserted-by":"crossref","unstructured":"P. Matus,\nThe maximum principle and some of its applications,\nComput. Methods Appl. Math. 2 (2002), no. 1, 50\u201391.","DOI":"10.2478\/cmam-2002-0004"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_016","doi-asserted-by":"crossref","unstructured":"P. Matus,\nOn convergence of difference schemes for IBVP for quasilinear parabolic equations with generalized solutions,\nComput. Methods Appl. Math. 14 (2014), no. 3, 361\u2013371.","DOI":"10.1515\/cmam-2014-0008"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_017","doi-asserted-by":"crossref","unstructured":"P. Matus, L. M. Hieu and L. G. Vulkov,\nAnalysis of second order difference schemes on non-uniform grids for quasilinear parabolic equations,\nJ. Comput. Appl. Math. 310 (2017), 186\u2013199.","DOI":"10.1016\/j.cam.2016.04.006"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_018","doi-asserted-by":"crossref","unstructured":"P. Matus and I. Rybak,\nDifference schemes for elliptic equations with mixed derivatives,\nComput. Methods Appl. Math. 4 (2004), no. 4, 494\u2013505.","DOI":"10.2478\/cmam-2004-0027"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_019","unstructured":"P. P. Matus,\nUnconditional convergence of some difference schemes of problems of gas dynamics,\nDiffer. Equ. 21 (1985), no. 7, 839\u2013848."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_020","doi-asserted-by":"crossref","unstructured":"P. P. Matus, L. M. Hieu and D. Pylak,\nMonotone finite-difference schemes of second-order accuracy for quasilinear parabolic equations with mixed derivatives (in Russian),\nDiffer. Uravn. 56 (2019), no. 3, 428\u2013440.","DOI":"10.1134\/S0012266119030157"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_021","doi-asserted-by":"crossref","unstructured":"P. P. Matus and D. B. Poliakov,\nConsistent two-sided estimates for the solutions of quasilinear parabolic equations and their approximations (in Russian),\nDiffer. Uravn. 53 (2017), no. 7, 991\u20131000.","DOI":"10.1134\/S0012266117070126"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_022","unstructured":"P. P. Matus and L. V. Stanishevskaya,\nUnconditional convergence of difference schemes for nonstationary quasilinear equations of mathematical physics,\nDiffer. Equ. 27 (1991), no. 7, 847\u2013859."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_023","unstructured":"P. P. Matus, V. T. K. Tuen and F. Z. Gaspar,\nMonotone difference schemes for a linear parabolic equation with boundary conditions of mixed type (in Russian),\nDokl. Nats. Akad. Nauk Belarusi 58 (2014), no. 5, 18\u201322."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_024","doi-asserted-by":"crossref","unstructured":"A. A. Samarskii,\nThe Theory of Difference Schemes,\nMonogr. Textb. Pure Appl. Math. 240,\nMarcel Dekker, New York, 2001.","DOI":"10.1201\/9780203908518"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_025","unstructured":"A. A. Samarskii, R. D. Lazarov and V. L. Makarov,\nDifference Schemes for Differential Equations with Generalized Solutions (in Russian),\nVysshaya Shkola, Moscow, 1987."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_026","doi-asserted-by":"crossref","unstructured":"A. A. Samarskii, P. P. Matus and P. N. Vabishchevich,\nDifference Schemes with Operator Factors,\nMath. Appl. 546,\nKluwer Academic, Dordrecht, 2002.","DOI":"10.1007\/978-94-015-9874-3"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_027","unstructured":"A. A. Samarski\u012d, V. I. Mazhukin, P. P. Matus and G. I. Shishkin,\nMonotone difference schemes for equations with mixed derivatives (in Russian),\nMat. Model. 13 (2001), no. 2, 17\u201326."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_028","unstructured":"A. A. Samarsk\u012d and P. N. Vabishchevich,\nNumerical Methods for Solution of Convection-Diffusion Problems (in Russian),\nEditorial YRSS, Moskow, 1999."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_029","unstructured":"A. A. Samarski\u012d, P. N. Vabishchevich and P. P. Matus,\nDifference schemes of increased order of accuracy on nonuniform grids (in Russian),\nDiffer. Uravn. 32 (1996), no. 2, 265\u2013274, 288."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_030","unstructured":"A. A. Samarski\u012d, P. N. Vabishchevich, A. N. Zyl and P. P. Matus,\nA difference scheme of the second order of accuracy for the Dirichlet problem in a domain of arbitrary shape (in Russian),\nMat. Model. 11 (1999), no. 9, 71\u201382."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_031","unstructured":"G. I. Shishkin,\nGrid approximation of a singularly perturbed boundary value problem for a quasilinear elliptic equation in the case of complete degeneration,\nComput. Math. Math. Phys. 31 (1991), no. 12, 33\u201346."},{"key":"2023033110450840980_j_cmam-2019-0052_ref_032","doi-asserted-by":"crossref","unstructured":"V. S. Vladimirov,\nEquations of mathematical physics (in Russian),\nNauka, Moscow, 1971.","DOI":"10.1063\/1.3022385"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_033","doi-asserted-by":"crossref","unstructured":"M. N. Yakovlev,\nSolvability of the finite-difference equations of the implicit scheme for a nonlinear second-order parabolic equation,\nJ. Sov. Math. 23 (1983), no. 1, 2081\u20132090.","DOI":"10.1007\/BF01093287"},{"key":"2023033110450840980_j_cmam-2019-0052_ref_034","doi-asserted-by":"crossref","unstructured":"M. N. Yakovlev,\nUniform convergence of the implicit scheme of the finite-difference method for solving the first boundary-value problem for a nonlinear second-order parabolic equation,\nJ. Sov. Math. 23 (1983), no. 1, 2066\u20132080.","DOI":"10.1007\/BF01093286"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/20\/4\/article-p695.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0052\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0052\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T12:57:06Z","timestamp":1680267426000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0052\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,19]]},"references-count":34,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,8,5]]},"published-print":{"date-parts":[[2020,10,1]]}},"alternative-id":["10.1515\/cmam-2019-0052"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2019-0052","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,7,19]]}}}