{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,4]],"date-time":"2026-10-04T01:35:13Z","timestamp":1791077713990,"version":"4.1.0"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","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 approximation of nonsmooth solutions of the transport equation in one space dimension by approximations given by a Runge\u2013Kutta discontinuous Galerkin method of order two. We take an initial datum, which has compact support and is smooth except at a discontinuity, and show that, if the ratio of the time step size to the grid size is less than $1\/3$, the error at the time T in the $L^2(\\mathbb{R}\\setminus\\mathcal{R}_T)$-norm is the optimal order two when $\\mathcal{R}_T$ is a region of size $O(T^{1\/2}\\,h^{1\/2}\\;\\log{1\/h})$ to the right of the discontinuity and of size $O(T^{1\/3}\\,h^{2\/3}\\;\\log{1\/h})$ to the left. Numerical experiments validating these results are presented.<\/jats:p>","DOI":"10.1137\/060668936","type":"journal-article","created":{"date-parts":[[2008,3,26]],"date-time":"2008-03-26T18:05:19Z","timestamp":1206554719000},"page":"1364-1398","source":"Crossref","is-referenced-by-count":26,"title":["Error Estimates for the Runge\u2013Kutta Discontinuous Galerkin Method for the Transport Equation with Discontinuous Initial Data"],"prefix":"10.1137","volume":"46","author":[{"given":"Bernardo","family":"Cockburn","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Johnny","family":"Guzm\u00e1n","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,3,26]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1968-0233527-6"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-11028"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-01-01317-5"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"B. Cockburn,\n                      An introduction to the discontinuous Galerkin method for convection-dominated problems\n                      , in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Math. 1697, Fond. C.I.M.E., A. Quarteroni, ed., Springer-Verlag, New York, 1998, pp. 151\u2013268.","DOI":"10.1007\/BFb0096353"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"B. Cockburn,\n                      Discontinuous Galerkin methods for convection-dominated problems\n                      , in High-Order Methods for Computational Physics, Lect. Notes Comput. Sci. Eng. 9, T. Barth and H. Deconink, eds., Springer-Verlag, Berlin, 1999, pp. 69\u2013224.","DOI":"10.1007\/978-3-662-03882-6_2"},{"key":"R6","first-page":"545","volume":"54","author":"Cockburn B.","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"B. 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Thom\u00e9e,\n                      Galerkin Finite Element Methods for Parabolic Equations\n                      , Springer-Verlag, New York, 1997.","DOI":"10.1007\/978-3-662-03359-3"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060668936","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:13:38Z","timestamp":1787325218000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060668936"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":20,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060668936"],"URL":"https:\/\/doi.org\/10.1137\/060668936","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}