{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,3]],"date-time":"2025-07-03T04:13:29Z","timestamp":1751516009512,"version":"3.41.0"},"reference-count":34,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071455","12471347"],"award-info":[{"award-number":["12071455","12471347"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This paper presents and analyzes energy stable local discontinuous Galerkin (LDG) methods for solving the cubic nonlinear shallow water wave equations, also referred to as the Camassa\u2013Holm\u2013Novikov (CHN) equations. These methods are characterized by their high-order accuracy and effectiveness in addressing the intricate nonlinear terms inherent to the CHN equations. The success of the proposed numerical schemes is due to the careful selection of the numerical fluxes at the cell boundaries, which are crucial for ensuring stability and accuracy. We employ a specific interpolation function to derive optimal a priori error estimates and provide a comprehensive analysis of the quadratic Taylor expansion of the cubic nonlinearity.\nIn addition, our numerical schemes show excellent performance for different peakon solutions. Numerical experiments validate the accuracy and efficiency of the proposed methods, demonstrating their effectiveness in practical applications.<\/jats:p>","DOI":"10.1515\/cmam-2024-0189","type":"journal-article","created":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T22:30:21Z","timestamp":1748644221000},"page":"665-694","source":"Crossref","is-referenced-by-count":1,"title":["High Order Energy Stable Local Discontinuous Galerkin Methods for Camassa\u2013Holm\u2013Novikov Equations"],"prefix":"10.1515","volume":"25","author":[{"given":"Jinyang","family":"Lu","sequence":"first","affiliation":[{"name":"School of Mathematics , Shandong University , Jinan , 250100 , P. R. China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6483-4336","authenticated-orcid":false,"given":"Yan","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences , 12652 University of Science and Technology of China , Hefei , Anhui 230026; and Laoshan Laboratory, Qingdao 266237 , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2025,5,29]]},"reference":[{"key":"2025070217063873285_j_cmam-2024-0189_ref_001","doi-asserted-by":"crossref","unstructured":"J. L.  Bona, H.  Chen, O.  Karakashian and Y.  Xing,\nConservative, discontinuous Galerkin-methods for the generalized Korteweg\u2013de Vries equation,\nMath. Comp. 82 (2013), no. 283, 1401\u20131432.","DOI":"10.1090\/S0025-5718-2013-02661-0"},{"key":"2025070217063873285_j_cmam-2024-0189_ref_002","doi-asserted-by":"crossref","unstructured":"R.  Camassa and D. D.  Holm,\nAn integrable shallow water equation with peaked solitons,\nPhys. Rev. Lett. 71 (1993), no. 11, 1661\u20131664.","DOI":"10.1103\/PhysRevLett.71.1661"},{"key":"2025070217063873285_j_cmam-2024-0189_ref_003","doi-asserted-by":"crossref","unstructured":"P. G.  Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nStud. Math. Appl. 4,\nNorth-Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"2025070217063873285_j_cmam-2024-0189_ref_004","doi-asserted-by":"crossref","unstructured":"B.  Cockburn, S.  Hou and C.-W.  Shu,\nThe Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case,\nMath. Comp. 54 (1990), no. 190, 545\u2013581.","DOI":"10.1090\/S0025-5718-1990-1010597-0"},{"key":"2025070217063873285_j_cmam-2024-0189_ref_005","doi-asserted-by":"crossref","unstructured":"B.  Cockburn, S. Y.  Lin and C.-W.  Shu,\nTVB Runge\u2013Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems,\nJ. Comput. Phys. 84 (1989), no. 1, 90\u2013113.","DOI":"10.1016\/0021-9991(89)90183-6"},{"key":"2025070217063873285_j_cmam-2024-0189_ref_006","doi-asserted-by":"crossref","unstructured":"B.  Cockburn and C.-W.  Shu,\nTVB Runge\u2013Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework,\nMath. Comp. 52 (1989), no. 186, 411\u2013435.","DOI":"10.1090\/S0025-5718-1989-0983311-4"},{"key":"2025070217063873285_j_cmam-2024-0189_ref_007","doi-asserted-by":"crossref","unstructured":"B.  Cockburn and C.-W.  Shu,\nThe Runge\u2013Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems,\nJ. Comput. Phys. 141 (1998), no. 2, 199\u2013224.","DOI":"10.1006\/jcph.1998.5892"},{"key":"2025070217063873285_j_cmam-2024-0189_ref_008","doi-asserted-by":"crossref","unstructured":"B.  Cockburn and C.-W.  Shu,\nRunge\u2013Kutta discontinuous Galerkin methods for convection-dominated problems,\nJ. Sci. 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