{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,4]],"date-time":"2026-10-04T06:48:26Z","timestamp":1791096506229,"version":"4.1.0"},"reference-count":39,"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 suggest a finite dfference scheme for the Camassa\u2013Holm equation that can handle general $H^1$ initial data. The form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality. We prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of the Camassa\u2013Holm equation.<\/jats:p>","DOI":"10.1137\/060673242","type":"journal-article","created":{"date-parts":[[2008,4,4]],"date-time":"2008-04-04T18:08:07Z","timestamp":1207332487000},"page":"1554-1579","source":"Crossref","is-referenced-by-count":41,"title":["A Convergent Finite Difference Scheme for the Camassa\u2013Holm Equation with General $H^1$ Initial Data"],"prefix":"10.1137","volume":"46","author":[{"given":"G. M.","family":"Coclite","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K. H.","family":"Karlsen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"N. H.","family":"Risebro","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,4,4]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2005.06.002"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1006\/aima.1999.1883"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1142\/S0219530507000857"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-006-0010-z"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.4310\/MAA.2005.v12.n2.a7"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.3934\/dcdsb.2003.3.115"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.71.1661"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/S0065-2156(08)70254-0"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.2991\/jnmp.2005.12.s1.13"},{"key":"R10","unstructured":"R. Camassa, J. Huang, and L. Lee,\n                      Integral and integrable algorithms for a nonlinear shallow-water wave equation\n                      , J. Comput. Phys., to appear."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/040616711"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2005.13.659"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2000.0701"},{"key":"R14","first-page":"303","volume":"26","author":"Constantin A.","year":"1998","journal-title":"Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.1998.47.1466"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392586"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/35\/32\/201"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/s00014-003-0785-6"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/s002200050801"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1007\/BF01170373"},{"key":"R22","doi-asserted-by":"crossref","first-page":"953","DOI":"10.57262\/die\/1356123175","volume":"14","author":"Danchin R.","year":"2001","journal-title":"Differential Integral Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0893-4983","issn-type":"print"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0396(03)00096-2"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"E. Feireisl,\n                      Dynamics of Viscous Compressible Fluids\n                      , Oxford Lecture Ser. Math. Appl.  26, Oxford University Press, Oxford, 2004.","DOI":"10.1093\/acprof:oso\/9780198528388.001.0001"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(00)00215-3"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"B. Fuchssteiner and A. S. Fokas,\n                      Symplectic structures, their B\u00e4cklund transformations and hereditary symmetries\n                      , Phys. D, 4 (1981\/82), pp. 47\u201366.","DOI":"10.1016\/0167-2789(81)90004-X"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1137\/040611975"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2006.14.505"},{"key":"R29","unstructured":"H. Holden and X. Raynaud,\n                      Global conservative solutions of the Camassa-Holm equation - A Lagrangian point of view\n                      , Comm. Partial Differential Equations, to appear."},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1006\/aima.1998.1721"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1137\/S1111111102410943"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112001007224"},{"key":"R33","unstructured":"H. Kalisch and X. Raynaud,\n                      Convergence of a spectral projection of the Camassa-Holm equation\n                      , Numer. Methods Partial Differential Equations, to appear."},{"key":"R34","doi-asserted-by":"crossref","unstructured":"R. J. LeVeque,\n                      Finite Volume Methods for Hyperbolic Problems\n                      , Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2002.","DOI":"10.1017\/CBO9780511791253"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1999.3683"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1016\/S0393-0440(97)00010-7"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1016\/S0362-546X(01)00791-X"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1002\/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1081\/PDE-120016129"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060673242","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:13:50Z","timestamp":1787325230000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060673242"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":39,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060673242"],"URL":"https:\/\/doi.org\/10.1137\/060673242","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}