{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T06:10:56Z","timestamp":1757311856178},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"2-3","license":[{"start":{"date-parts":[[2023,2,15]],"date-time":"2023-02-15T00:00:00Z","timestamp":1676419200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,2,15]],"date-time":"2023-02-15T00:00:00Z","timestamp":1676419200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Institute of Mathematics of the Czech Academy of Sciences"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We present new error estimates for the finite volume and finite difference methods applied to the compressible Navier\u2013Stokes equations. The main innovative ingredients of the improved error estimates are a refined consistency analysis combined with a continuous version of the relative energy inequality. Consequently, we obtain better convergence rates than those available in the literature so far. Moreover, the error estimates hold in the whole physically relevant range of the adiabatic coefficient.<\/jats:p>","DOI":"10.1007\/s00211-023-01346-y","type":"journal-article","created":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T07:33:16Z","timestamp":1676619196000},"page":"493-529","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Improved error estimates for the finite volume and the MAC schemes for the compressible Navier\u2013Stokes system"],"prefix":"10.1007","volume":"153","author":[{"given":"Eduard","family":"Feireisl","sequence":"first","affiliation":[]},{"given":"M\u00e1ria","family":"Luk\u00e1\u010dov\u00e1-Medvid\u2019ov\u00e1","sequence":"additional","affiliation":[]},{"given":"Bangwei","family":"She","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,2,15]]},"reference":[{"issue":"2","key":"1346_CR1","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1080\/03605302.2018.1442476","volume":"43","author":"D Breit","year":"2018","unstructured":"Breit, D., Feireisl, E., Hofmanov\u00e1, M.: Local strong solutions to the stochastic compressible Navier\u2013Stokes system. Commun. Partial Differ. Equ. 43(2), 313\u2013345 (2018)","journal-title":"Commun. Partial Differ. Equ."},{"key":"1346_CR2","series-title":"Springer Series in Computational Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-19267-3","volume-title":"Discontinuous Galerkin Method","author":"V Dolej\u0161\u00ed","year":"2015","unstructured":"Dolej\u0161\u00ed, V., Feistauer, M.: Discontinuous Galerkin Method. Springer Series in Computational Mathematics, vol. 48. Springer (2015)"},{"key":"1346_CR3","unstructured":"Feistauer, M.: Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 67. Longman Scientific & Technical, Harlow (1993)"},{"key":"1346_CR4","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198505884.001.0001","volume-title":"Mathematical and Computational Methods for Compressible Flow","author":"M Feistauer","year":"2003","unstructured":"Feistauer, M., Felcman, J., Stra\u0161kraba, I.: Mathematical and Computational Methods for Compressible Flow. The Clarendon Press, Oxford University Press (2003)"},{"key":"1346_CR5","first-page":"713","volume":"7","author":"R Eymard","year":"2000","unstructured":"Eymard, R., Gallou\u00ebt, T., Herbin, R.: Finite volume methods. Handb. Numer. Anal. 7, 713\u20131018 (2000)","journal-title":"Handb. Numer. Anal."},{"issue":"1","key":"1346_CR6","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1051\/m2an\/2016022","volume":"51","author":"E Feireisl","year":"2017","unstructured":"Feireisl, E., Ho\u0161ek, R., Maltese, D., Novotn\u00fd, A.: Error estimates for a numerical method for the compressible Navier\u2013Stokes system on sufficiently smooth domains. ESAIM Math. Model. Numer. Anal. 51(1), 279\u2013319 (2017)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"1346_CR7","series-title":"Oxford Lecture Series in Mathematics and its Applications","volume-title":"Dynamics of Viscous Compressible Fluids","author":"E Feireisl","year":"2004","unstructured":"Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and its Applications, Oxford University Press (2004)"},{"issue":"4","key":"1346_CR8","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1007\/s00021-011-0091-9","volume":"14","author":"E Feireisl","year":"2012","unstructured":"Feireisl, E., Jin, B.J., Novotn\u00fd, A.: Relative entropies, suitable weak solutions, and weak strong uniqueness for the compressible Navier\u2013Stokes system. J. Math. Fluid Mech. 14(4), 717\u2013730 (2012)","journal-title":"J. Math. Fluid Mech."},{"key":"1346_CR9","volume-title":"Mathematical Theory of Compressible Viscous Fluids: Analysis and Numerics","author":"E Feireisl","year":"2017","unstructured":"Feireisl, E., Karper, T.G., Pokorn\u00fd, M.: Mathematical Theory of Compressible Viscous Fluids: Analysis and Numerics. Birkh\u00e4user-Verlag, Basel (2017)"},{"issue":"3","key":"1346_CR10","doi-asserted-by":"publisher","first-page":"703","DOI":"10.1007\/s10208-017-9351-2","volume":"18","author":"E Feireisl","year":"2018","unstructured":"Feireisl, E., Luk\u00e1\u010dov\u00e1-Medvid\u2019ov\u00e1, M.: Convergence of a mixed finite element-finite volume scheme for the isentropic Navier\u2013Stokes system via the dissipative measure-valued solutions. Found. Comput. Math. 18(3), 703\u2013730 (2018)","journal-title":"Found. Comput. Math."},{"issue":"6","key":"1346_CR11","doi-asserted-by":"publisher","first-page":"1957","DOI":"10.1051\/m2an\/2019043","volume":"53","author":"E Feireisl","year":"2019","unstructured":"Feireisl, E., Luk\u00e1\u010dov\u00e1-Medvid\u2019ov\u00e1, M., Mizerov\u00e1, H., She, B.: Convergence of a finite volume scheme for the compressible Navier\u2013Stokes system. ESAIM: M2AN 53(6), 1957\u20131979 (2019)","journal-title":"ESAIM: M2AN"},{"key":"1346_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-73788-7","volume-title":"Numerical Analysis of Compressible Fluid Flows","author":"E Feireisl","year":"2021","unstructured":"Feireisl, E., Luk\u00e1\u010dov\u00e1-Medvid\u2019ov\u00e1, M., Mizerov\u00e1, H., She, B.: Numerical Analysis of Compressible Fluid Flows. Springer (2021)"},{"issue":"3","key":"1346_CR13","first-page":"525","volume":"40","author":"I Gallagher","year":"2000","unstructured":"Gallagher, I.: A remark on smooth solutions of the weakly compressible periodic Navier\u2013Stokes equations. J. Math. Kyoto Univ. 40(3), 525\u2013540 (2000)","journal-title":"J. Math. Kyoto Univ."},{"key":"1346_CR14","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/s00211-018-1007-x","volume":"141","author":"T Gallou\u00ebt","year":"2019","unstructured":"Gallou\u00ebt, T., Maltese, D., Novotn\u00fd, A.: Error estimates for the implicit MAC scheme for the compressible Navier\u2013Stokes equations. Numer. Math. 141, 495\u2013567 (2019)","journal-title":"Numer. Math."},{"issue":"2","key":"1346_CR15","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1093\/imanum\/drv028","volume":"36","author":"T Gallou\u00ebt","year":"2016","unstructured":"Gallou\u00ebt, T., Herbin, R., Maltese, D., Novotn\u00fd, A.: Error estimates for a numerical approximation to the compressible barotropic Navier\u2013Stokes equations. IMA J. Numer. Anal. 36(2), 543\u2013592 (2016)","journal-title":"IMA J. Numer. Anal."},{"issue":"3","key":"1346_CR16","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1515\/jnma-2017-0010","volume":"26","author":"R Ho\u0161ek","year":"2018","unstructured":"Ho\u0161ek, R., She, B.: Stability and consistency of a finite difference scheme for compressible viscous isentropic flow in multi-dimension. J. Numer. Math. 26(3), 111\u2013140 (2018)","journal-title":"J. Numer. Math."},{"key":"1346_CR17","first-page":"263","volume":"30","author":"V Jovanovi\u0107","year":"2007","unstructured":"Jovanovi\u0107, V.: An error estimate for a numerical scheme for the compressible Navier\u2013Stokes system. Kragujev. J. Math. 30, 263\u2013275 (2007)","journal-title":"Kragujev. J. Math."},{"issue":"3","key":"1346_CR18","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1007\/s00211-013-0543-7","volume":"125","author":"T Karper","year":"2013","unstructured":"Karper, T.: A convergent FEM-DG method for the compressible Navier\u2013Stokes equations. Numer. Math. 125(3), 441\u2013510 (2013)","journal-title":"Numer. Math."},{"issue":"1","key":"1346_CR19","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1093\/imanum\/draa093","volume":"42","author":"Y Kwon","year":"2022","unstructured":"Kwon, Y., Novotn\u00fd, A.: Consistency, convergence and error estimates for a mixed finite element-finite volume scheme to compressible Navier\u2013Stokes equations with general inflow\/outflow boundary data. IMA J. Numer. Anal. 42(1), 107\u2013164 (2022)","journal-title":"IMA J. Numer. Anal."},{"key":"1346_CR20","volume-title":"Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models","author":"PL Lions","year":"1998","unstructured":"Lions, P.L.: Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models. Oxford University Press (1998)"},{"key":"1346_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/S0036142998336424","volume":"38","author":"B Liu","year":"2000","unstructured":"Liu, B.: The analysis of a finite element method with streamline diffusion for the compressible Navier\u2013Stokes equations. SIAM J. Numer. Anal. 38, 1\u201316 (2000)","journal-title":"SIAM J. Numer. Anal."},{"key":"1346_CR22","doi-asserted-by":"publisher","first-page":"432","DOI":"10.1002\/num.10102","volume":"20","author":"B Liu","year":"2004","unstructured":"Liu, B.: On a finite element method for three-dimensional unsteady compressible viscous flows. Numer. Methods Partial Differ. Equ. 20, 432\u2013449 (2004)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"1346_CR23","unstructured":"Luk\u00e1\u010dov\u00e1-Medvid\u2019ov\u00e1, M.,\u00a0She, B., Yuan, Y.: Convergence and error estimates of a penalization finite volume method for the compressible Navier\u2013Stokes system. arXiv preprint arXiv:2209.02344"},{"issue":"1","key":"1346_CR24","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1007\/s10915-020-01278-x","volume":"84","author":"H Mizerov\u00e1","year":"2020","unstructured":"Mizerov\u00e1, H., She, B.: Convergence and error estimates for a finite difference scheme for the multi-dimensional compressible Navier\u2013Stokes system. J. Sci. Comput. 84(1), 25 (2020)","journal-title":"J. Sci. Comput."},{"issue":"1","key":"1346_CR25","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1016\/j.matpur.2010.08.001","volume":"95","author":"Y Sun","year":"2011","unstructured":"Sun, Y., Wang, C., Zhang, Z.: A Beale\u2013Kato\u2013Majda blow-up criterion for the 3-D compressible Navier\u2013Stokes equations. J. Math. Pures. Appl. 95(1), 36\u201347 (2011)","journal-title":"J. Math. Pures. Appl."},{"key":"1346_CR26","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/BF01206939","volume":"103","author":"A Valli","year":"1986","unstructured":"Valli, A., Zajaczkowski, M.: Navier\u2013Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103, 259\u2013296 (1986)","journal-title":"Commun. Math. Phys."},{"issue":"1","key":"1346_CR27","doi-asserted-by":"publisher","first-page":"626","DOI":"10.1137\/140960542","volume":"47","author":"PI Plotnikov","year":"2015","unstructured":"Plotnikov, P.I., Weigant, W.: Isothermal Navier\u2013Stokes equations and Radon transform. SIAM J. Math. Anal. 47(1), 626\u2013653 (2015)","journal-title":"SIAM J. Math. Anal."},{"key":"1346_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/b79761","volume-title":"Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction","author":"EF Toro","year":"2009","unstructured":"Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Springer (2009)"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-023-01346-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-023-01346-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-023-01346-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,7]],"date-time":"2023-12-07T01:46:41Z","timestamp":1701913601000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-023-01346-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,15]]},"references-count":28,"journal-issue":{"issue":"2-3","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["1346"],"URL":"https:\/\/doi.org\/10.1007\/s00211-023-01346-y","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,15]]},"assertion":[{"value":"9 May 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 January 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 February 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}