{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,4]],"date-time":"2026-10-04T17:46:14Z","timestamp":1791135974212,"version":"4.1.0"},"reference-count":34,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>In this paper, we develop, analyze, and test a local discontinuous Galerkin (LDG) method for solving the Camassa\u2013Holm equation which contains nonlinear high-order derivatives. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the $L^2$ stability for general solutions and give a detailed error estimate for smooth solutions, and provide numerical simulation results for different types of solutions of the nonlinear Camassa\u2013Holm equation to illustrate the accuracy and capability of the LDG method.<\/jats:p>","DOI":"10.1137\/070679764","type":"journal-article","created":{"date-parts":[[2008,4,23]],"date-time":"2008-04-23T18:23:16Z","timestamp":1208974996000},"page":"1998-2021","source":"Crossref","is-referenced-by-count":127,"title":["A Local Discontinuous Galerkin Method for the Camassa\u2013Holm Equation"],"prefix":"10.1137","volume":"46","author":[{"given":"Yan","family":"Xu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chi-Wang","family":"Shu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,4,23]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2005.06.002"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-006-0010-z"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1142\/S0219530507000857"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.71.1661"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/S0065-2156(08)70254-0"},{"key":"R6","unstructured":"P. Ciarlet,\n                      The Finite Element Method for Elliptic Problem\n                      , North\u2013Holland, Amsterdam, 1975."},{"key":"R7","doi-asserted-by":"crossref","unstructured":"B. Cockburn,\n                      Discontinuous Galerkin methods for methods for convection-dominated problems\n                      , in High-Order Methods for Computational Physics, T. J. Barth and H. Deconinck, eds., Lecture Notes in Computational Science and Engineering 9, Springer, 1999, pp. 69\u2013224.","DOI":"10.1007\/978-3-662-03882-6_2"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1989-0983311-4"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142997316712"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1023\/A:1012873910884"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-004-4131-y"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2005.06.010"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1023\/A:1015144429543"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-004-4140-x"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2005.07.017"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2006.14.505"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/040611975"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112001007224"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2002.1078"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2004.11.024"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1002\/num.20140"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2004.09.007"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/38\/4\/007"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.11.013"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2004.1331"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(88)90177-5"},{"key":"R27","first-page":"250","volume":"22","author":"Xu Y.","year":"2004","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.11.001"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2005.06.007"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2005.06.021"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2006.10.043"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901390378"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1023\/A:1015132126817"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902404182"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070679764","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:14:14Z","timestamp":1787325254000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070679764"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":34,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/070679764"],"URL":"https:\/\/doi.org\/10.1137\/070679764","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}