{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T08:03:41Z","timestamp":1791273821260,"version":"4.1.0"},"reference-count":16,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We study the convergence of the upwind finite volume scheme applied to the linear advection equation with a Lipschitz divergence-free speed in $\\R^d$. We prove an $h^{1\/2-\\varepsilon}$-error estimate in the $L^\\infty(\\R^d\\times [0,T])$-norm for Lipschitz initial data. The expected optimal result is an $h^{1\/2}$-error estimate. In a second part, we also prove an $h^{1\/2}$-error estimate in the $L^\\infty(0,T;L^2(\\R^d))$-norm for initial data in $H^1(\\R^d)$.<\/jats:p>","DOI":"10.1137\/060664057","type":"journal-article","created":{"date-parts":[[2007,12,20]],"date-time":"2007-12-20T18:32:01Z","timestamp":1198175521000},"page":"124-150","source":"Crossref","is-referenced-by-count":26,"title":["$L^\\infty$- and $L^2$-Error Estimates for a Finite Volume Approximation of Linear Advection"],"prefix":"10.1137","volume":"46","author":[{"given":"Benoit","family":"Merlet","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,12,20]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/040605941"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1994-1240657-4"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142997316165"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1980-0553381-X"},{"key":"R5","unstructured":"B. Despr\u00e9s,\n                      Convergence of non-linear finite volume schemes for linear transport\n                      , in Notes from the XIth Jacques-Louis Lions Hispano-French School on Numerical Simulation in Physics and Engineering (Spanish), Grupo Anal. Teor. Numer. Modelos Cienc. Exp. Univ. C\u00e1diz, C\u00e1diz, Spain, 2004, pp. 219\u2013239."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-03-01618-1"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"R. Eymard, T. Gallou\u00ebt, and R. Herbin,\n                      Finite volume methods\n                      , in Handbook of Numerical Analysis, Vol. VII, Handb. Numer. Anal. VII, North-Holland, Amsterdam, 2000, pp. 713\u20131020.","DOI":"10.1016\/S1570-8659(00)07005-8"},{"key":"R8","doi-asserted-by":"crossref","first-page":"419","DOI":"10.57262\/ade\/1356651802","volume":"7","author":"Eymard R.","year":"2002","journal-title":"Adv. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/1079-9389","issn-type":"print"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1002\/num.1690110205"},{"key":"R10","unstructured":"D. Kr\u00f6ner and M. Rokyta,\n                      Higher order finite volume method for convection dominated diffusion equation in 2D\n                      , in Numerical Modelling in Continuum Mechanics, M. Feistauer, R. Rannacher, K. Kozel, eds., Matfyzpress, Praha, Czech Republic, 1997, pp. 62\u201370."},{"key":"R11","first-page":"1489","volume":"16","author":"Kuznecov N. N.","year":"1976","journal-title":"\u017d. Vy\u010disl. Mat. i Mat. Fiz."},{"key":"R12","unstructured":"B. Merlet and J. Vovelle,\n                      Error estimate for a finite volume scheme applied to the linear advection equation\n                      , Numer. Math., to appear."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142997328068"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/22.2.253"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1994280302671"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-002-0396-y"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060664057","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:09:17Z","timestamp":1787324957000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060664057"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,12,20]]},"references-count":16,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060664057"],"URL":"https:\/\/doi.org\/10.1137\/060664057","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,12,20]]}}}