{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T04:24:02Z","timestamp":1777091042969,"version":"3.51.4"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2014,6,8]],"date-time":"2014-06-08T00:00:00Z","timestamp":1402185600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2015,2]]},"DOI":"10.1007\/s00211-014-0641-1","type":"journal-article","created":{"date-parts":[[2014,6,10]],"date-time":"2014-06-10T17:44:21Z","timestamp":1402422261000},"page":"321-351","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["A local discontinuous Galerkin method for the Burgers\u2013Poisson equation"],"prefix":"10.1007","volume":"129","author":[{"given":"Hailiang","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nattapol","family":"Ploymaklam","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2014,6,8]]},"reference":[{"issue":"3","key":"641_CR1","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1016\/j.jde.2013.04.016","volume":"255","author":"D Amadori","year":"2013","unstructured":"Amadori, D., Gosse, L.: Transient $$L^1$$ L 1 error estimates for well-balanced schemes on non-resonant scalar balance laws. J. Differ. Equ. 255(3), 469\u2013502 (2013)","journal-title":"J. Differ. Equ."},{"issue":"283","key":"641_CR2","doi-asserted-by":"crossref","first-page":"1401","DOI":"10.1090\/S0025-5718-2013-02661-0","volume":"82","author":"JL Bona","year":"2013","unstructured":"Bona, J.L., Chen, H., Karakashian, O.A., Xing, Y.: Conservative, discontinuous Galerkin methods for the generalized Korteweg-de Vries equation. Math. Comput. 82(283), 1401\u20131432 (2013)","journal-title":"Math. Comput."},{"issue":"1695","key":"641_CR3","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1098\/rsta.1995.0027","volume":"351","author":"JL Bona","year":"1995","unstructured":"Bona, J.L., Dougalis, V.A., Karakashian, O.A., McKinney, W.R.: Conservative, high-order numerical schemes for the generalized Korteweg\u2013de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A 351(1695), 107\u2013164 (1995)","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A"},{"issue":"11","key":"641_CR4","doi-asserted-by":"crossref","first-page":"1661","DOI":"10.1103\/PhysRevLett.71.1661","volume":"71","author":"R Camassa","year":"1993","unstructured":"Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661\u20131664 (1993)","journal-title":"Phys. Rev. Lett."},{"issue":"31","key":"641_CR5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0065-2156(08)70254-0","volume":"31","author":"R Camassa","year":"1994","unstructured":"Camassa, R., Holm, D.D., Hyman, J.M.: A new integrable shallow water equation. Adv. Appl. Mech. 31(31), 1\u201333 (1994)","journal-title":"Adv. Appl. Mech."},{"issue":"238","key":"641_CR6","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1090\/S0025-5718-01-01317-5","volume":"71","author":"P Castillo","year":"2002","unstructured":"Castillo, P., Cockburn, B., Sch\u00f6tzau, D., Schwab, C.: Optimal a priori error estimates for the $$hp$$ h p -version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71(238), 455\u2013478 (2002)","journal-title":"Math. Comput."},{"issue":"20","key":"641_CR7","doi-asserted-by":"crossref","first-page":"6770","DOI":"10.1016\/j.jcp.2012.06.022","volume":"231","author":"E Celledoni","year":"2012","unstructured":"Celledoni, E., Grimm, V., McLachlan, R.I., McLaren, D.I., O\u2019Neale, D., Owren, B., Quispel, G.R.W.: Preserving energy resp. dissipation in numerical PDEs using the \u201caverage vector field\u201d method. J. Comput. Phys. 231(20), 6770\u20136789 (2012)","journal-title":"J. Comput. Phys."},{"key":"641_CR8","volume-title":"The Finite Element Method for Elliptic Problems","author":"PG Ciarlet","year":"1978","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, New York (1978)"},{"issue":"190","key":"641_CR9","first-page":"545","volume":"54","author":"B Cockburn","year":"1990","unstructured":"Cockburn, B., Hou, S., Shu, C.-W.: The Runge\u2013Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54(190), 545\u2013581 (1990)","journal-title":"Math. Comput."},{"issue":"1","key":"641_CR10","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/0021-9991(89)90183-6","volume":"84","author":"B Cockburn","year":"1989","unstructured":"Cockburn, B., Lin, S.Y., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84(1), 90\u2013113 (1989)","journal-title":"J. Comput. Phys."},{"issue":"186","key":"641_CR11","first-page":"411","volume":"52","author":"B Cockburn","year":"1989","unstructured":"Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411\u2013435 (1989)","journal-title":"Math. Comput."},{"issue":"6","key":"641_CR12","doi-asserted-by":"crossref","first-page":"2440","DOI":"10.1137\/S0036142997316712","volume":"35","author":"B Cockburn","year":"1998","unstructured":"Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440\u20132463 (1998). (electronic)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"641_CR13","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1006\/jcph.1998.5892","volume":"141","author":"B Cockburn","year":"1998","unstructured":"Cockburn, B.: The Runge\u2013Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems. J. Comput. Phys. 141(2), 199\u2013224 (1998)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"641_CR14","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s00205-008-0128-2","volume":"192","author":"A Constantin","year":"2009","unstructured":"Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa\u2013Holm and Degasperis\u2013Procesi equations. Arch. Ration. Mech. Anal. 192(1), 165\u2013186 (2009)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"2","key":"641_CR15","doi-asserted-by":"crossref","first-page":"170","DOI":"10.4213\/tmf388","volume":"133","author":"A Degasperis","year":"2002","unstructured":"Degasperis, A., Kholm, D.D., Khon, A.N.I.: A new integrable equation with peakon solutions. Teoret. Mat. Fiz. 133(2), 170\u2013183 (2002)","journal-title":"Teoret. Mat. Fiz."},{"issue":"5","key":"641_CR16","doi-asserted-by":"crossref","first-page":"1509","DOI":"10.1137\/S0036139902410345","volume":"64","author":"K Fellner","year":"2004","unstructured":"Fellner, K., Schmeiser, C.: Burgers-Poisson: a nonlinear dispersive model equation. SIAM J. Appl. Math. 64(5), 1509\u20131525 (2004). (electronic)","journal-title":"SIAM J. Appl. Math."},{"issue":"1","key":"641_CR17","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/0167-2789(81)90004-X","volume":"4","author":"B Fuchssteiner","year":"1981\/1982","unstructured":"Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their B\u00e4cklund transformations and hereditary symmetries. Phys. D 4(1), 47\u201366 (1981\/1982)","journal-title":"Phys. D"},{"key":"641_CR18","first-page":"317","volume":"8","author":"D Furihata","year":"1998","unstructured":"Furihata, D., Mori, M.: General derivation of finite difference schemes by means of a discrete variation. Trans. Japan Soc. Ind. Appl. Math. 8, 317\u2013340 (1998)","journal-title":"Trans. Japan Soc. Ind. Appl. Math."},{"issue":"221","key":"641_CR19","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1090\/S0025-5718-98-00913-2","volume":"67","author":"S Gottlieb","year":"1998","unstructured":"Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge\u2013Kutta schemes. Math. Comput. 67(221), 73\u201385 (1998)","journal-title":"Math. Comput."},{"issue":"18","key":"641_CR20","doi-asserted-by":"crossref","first-page":"1192","DOI":"10.1103\/PhysRevLett.27.1192","volume":"27","author":"R Hirota","year":"1971","unstructured":"Hirota, R.: Exact solution of the Korteweg\u2013de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27(18), 1192\u20131194 (1971)","journal-title":"Phys. Rev. Lett."},{"issue":"3","key":"641_CR21","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1137\/S1111111102410943","volume":"2","author":"DD Holm","year":"2003","unstructured":"Holm, D.D., Staley, M.F.: Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dyn. Syst. 2(3), 323\u2013380 (2003). (electronic)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"641_CR22","doi-asserted-by":"crossref","unstructured":"Johnson, R.S., Camassa-Holm.: Korteweg\u2013de Vries and related models for water waves. J. Fluid Mech. 455, 63\u201382 (2002)","DOI":"10.1017\/S0022112001007224"},{"key":"641_CR23","doi-asserted-by":"crossref","unstructured":"Liu, H.: Optimal error estimates of the direct discontinuous Galerkin method for convection\u2013diffusion equations. Math. Comput. (2014, accepted)","DOI":"10.1090\/S0025-5718-2015-02923-8"},{"issue":"1","key":"641_CR24","first-page":"83","volume":"21","author":"H Liu","year":"2014","unstructured":"Liu, H., Huang, Y., Yi, N.: A direct discontinuous Galerkin method for the Degasperis\u2013Procesi equation. Methods Appl. Anal. 21(1), 83\u2013106 (2014)","journal-title":"Methods Appl. Anal."},{"issue":"1","key":"641_CR25","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.jcp.2005.10.016","volume":"215","author":"H Liu","year":"2006","unstructured":"Liu, H., Yan, J.: A local discontinuous Galerkin method for the Korteweg\u2013de Vries equation with boundary effect. J. Comput. Phys. 215(1), 197\u2013218 (2006)","journal-title":"J. Comput. Phys."},{"key":"641_CR26","doi-asserted-by":"crossref","unstructured":"Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal. 47(1), 675\u2013698 (2008\/2009)","DOI":"10.1137\/080720255"},{"issue":"3","key":"641_CR27","doi-asserted-by":"crossref","first-page":"541","DOI":"10.4208\/cicp.010909.011209a","volume":"8","author":"H Liu","year":"2010","unstructured":"Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 8(3), 541\u2013564 (2010)","journal-title":"Commun. Comput. Phys."},{"key":"641_CR28","doi-asserted-by":"crossref","unstructured":"Liu, H., Yin, Z.: Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations. In Nonlinear partial differential equations and hyperbolic wave phenomena, Contemp. Math., vol. 526, pp. 273\u2013294. Amer. Math. Soc., Providence (2010)","DOI":"10.1090\/conm\/526\/10385"},{"issue":"3","key":"641_CR29","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1007\/s00220-006-0082-5","volume":"267","author":"Y Liu","year":"2006","unstructured":"Liu, Y., Yin, Z.: Global existence and blow-up phenomena for the Degasperis\u2013Procesi equation. Commun. Math. Phys. 267(3), 801\u2013820 (2006)","journal-title":"Commun. Math. Phys."},{"issue":"2","key":"641_CR30","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1016\/j.cam.2007.08.001","volume":"218","author":"T Matsuo","year":"2008","unstructured":"Matsuo, T.: Dissipative\/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations. J. Comput. Appl. Math. 218(2), 506\u2013521 (2008)","journal-title":"J. Comput. Appl. Math."},{"issue":"1754","key":"641_CR31","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1098\/rsta.1999.0363","volume":"357","author":"RI McLachlan","year":"1999","unstructured":"McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357(1754), 1021\u20131045 (1999)","journal-title":"R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci."},{"key":"641_CR32","unstructured":"Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Los Alamos, Report LA-UR-73-479 (1973)"},{"issue":"25","key":"641_CR33","doi-asserted-by":"crossref","first-page":"2765","DOI":"10.1016\/S0045-7825(03)00294-9","volume":"192","author":"T Warburton","year":"2003","unstructured":"Warburton, T., Hesthaven, J.S.: On the constants in $$hp$$ h p -finite element trace inverse inequalities. Comput. Methods Appl. Mech. Eng. 192(25), 2765\u20132773 (2003)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"641_CR34","volume-title":"Linear and Nonlinear Waves","author":"GB Whitham","year":"1974","unstructured":"Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)"},{"issue":"4","key":"641_CR35","doi-asserted-by":"crossref","first-page":"1998","DOI":"10.1137\/070679764","volume":"46","author":"Y Xu","year":"2008","unstructured":"Xu, Y., Shu, C.-W.: A local discontinuous Galerkin method for the Camassa\u2013Holm equation. SIAM J. Numer. Anal. 46(4), 1998\u20132021 (2008)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"641_CR36","first-page":"1","volume":"7","author":"Y Xu","year":"2010","unstructured":"Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7(1), 1\u201346 (2010)","journal-title":"Commun. Comput. Phys."},{"issue":"1\u20132","key":"641_CR37","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.physd.2005.06.007","volume":"208","author":"Y Xu","year":"2005","unstructured":"Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations. Phys. D Nonlinear Phenom. 208(1\u20132), 21\u201358 (2005)","journal-title":"Phys. D Nonlinear Phenom."},{"issue":"37\u201340","key":"641_CR38","doi-asserted-by":"crossref","first-page":"3805","DOI":"10.1016\/j.cma.2006.10.043","volume":"196","author":"Y Xu","year":"2007","unstructured":"Xu, Y., Shu, C.W.: Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection\u2013diffusion and KdV equations. Comput. Methods Appl. Mech. Eng. 196(37\u201340), 3805\u20133822 (2007)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"641_CR39","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1137\/S0036142901390378","volume":"40","author":"J Yan","year":"2003","unstructured":"Yan, J., Shu, C.W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40, 769\u2013791 (2003)","journal-title":"SIAM J. Numer. Anal."},{"key":"641_CR40","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/j.jcp.2013.01.031","volume":"242","author":"N Yi","year":"2013","unstructured":"Yi, N., Huang, Y., Liu, H.: A direct discontinuous Galerkin method for the generalized Korteweg\u2013de Vries equation: energy conservation and boundary effect. J. Comput. Phys. 242, 351\u2013366 (2013)","journal-title":"J. Comput. Phys."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-014-0641-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-014-0641-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-014-0641-1","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,7]],"date-time":"2022-04-07T13:42:49Z","timestamp":1649338969000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-014-0641-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,8]]},"references-count":40,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2015,2]]}},"alternative-id":["641"],"URL":"https:\/\/doi.org\/10.1007\/s00211-014-0641-1","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,6,8]]}}}