{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,5]],"date-time":"2025-03-05T05:16:47Z","timestamp":1741151807694,"version":"3.38.0"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,11,23]],"date-time":"2024-11-23T00:00:00Z","timestamp":1732320000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,11,23]],"date-time":"2024-11-23T00:00:00Z","timestamp":1732320000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100005046","name":"Natural Science Foundation of Heilongjiang Province","doi-asserted-by":"publisher","award":["LH2020A015"],"award-info":[{"award-number":["LH2020A015"]}],"id":[{"id":"10.13039\/501100005046","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,2]]},"DOI":"10.1007\/s40314-024-03002-z","type":"journal-article","created":{"date-parts":[[2024,11,23]],"date-time":"2024-11-23T13:10:24Z","timestamp":1732367424000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bound-preserving schemes for $$P^2$$ local discontinuous Galerkin discretizations of KdV-type equations"],"prefix":"10.1007","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5062-137X","authenticated-orcid":false,"given":"Hui","family":"Bi","sequence":"first","affiliation":[]},{"given":"Feilong","family":"Zhao","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,11,23]]},"reference":[{"key":"3002_CR1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511998324","volume-title":"Nonlinear dispersive waves: asymptotic analysis and solitons. Cambridge texts in applied mathematics","author":"MJ Ablowitz","year":"2011","unstructured":"Ablowitz MJ (2011) Nonlinear dispersive waves: asymptotic analysis and solitons. Cambridge texts in applied mathematics. Cambridge University Press, Cambridge"},{"issue":"7","key":"3002_CR2","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1016\/j.physleta.2012.12.040","volume":"377","author":"MJ Ablowitz","year":"2013","unstructured":"Ablowitz MJ, Baldwin DE (2013) Interactions and asymptotics of dispersive shock waves\u2013Korteweg\u2013de Vries equation. Phys Lett A 377(7):555\u2013559","journal-title":"Phys Lett A"},{"key":"3002_CR3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.80.016603","volume":"80","author":"MJ Ablowitz","year":"2009","unstructured":"Ablowitz MJ, Baldwin DE, Hoefer MA (2009) Soliton generation and multiple phases in dispersive shock and rarefaction wave interaction. Phys Rev E 80:016603","journal-title":"Phys Rev E"},{"issue":"283","key":"3002_CR4","doi-asserted-by":"publisher","first-page":"1401","DOI":"10.1090\/S0025-5718-2013-02661-0","volume":"82","author":"J Bona","year":"2013","unstructured":"Bona J, Chen H, Karakashian O, Xing Y (2013) Conservative, discontinuous Galerkin-methods for the generalized Korteweg\u2013de Vries equation. Math Comput 82(283):1401\u20131432","journal-title":"Math Comput"},{"issue":"262","key":"3002_CR5","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1090\/S0025-5718-07-02045-5","volume":"77","author":"Y Cheng","year":"2008","unstructured":"Cheng Y, Shu CW (2008) A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Math Comput 77(262):699\u2013730","journal-title":"Math Comput"},{"key":"3002_CR6","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1016\/j.jcp.2018.11.003","volume":"378","author":"N Chuenjarern","year":"2019","unstructured":"Chuenjarern N, Xu Z, Yang Y (2019) High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes. J Comput Phys 378:110\u2013128","journal-title":"J Comput Phys"},{"key":"3002_CR7","volume-title":"Hyperbolic conservation laws in continuum physics","author":"CM Dafermos","year":"2009","unstructured":"Dafermos CM (2009) Hyperbolic conservation laws in continuum physics. Springer, Berlin"},{"issue":"2","key":"3002_CR8","doi-asserted-by":"publisher","first-page":"200","DOI":"10.1016\/S0167-2789(99)00072-X","volume":"134","author":"A Debussche","year":"1999","unstructured":"Debussche A, Printems J (1999) Numerical simulation of the stochastic Korteweg\u2013de Vries equation. Physica D 134(2):200\u2013226","journal-title":"Physica D"},{"key":"3002_CR9","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1016\/j.jcp.2018.10.034","volume":"377","author":"J Du","year":"2019","unstructured":"Du J, Yang Y (2019) Maximum-principle-preserving third-order local discontinuous Galerkin method for convection\u2013diffusion equations on overlapping meshes. J Comput Phys 377:117\u2013141","journal-title":"J Comput Phys"},{"key":"3002_CR10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111548","volume":"469","author":"J Du","year":"2022","unstructured":"Du J, Yang Y (2022) High-order bound-preserving discontinuous Galerkin methods for multicomponent chemically reacting flows. J Comput Phys 469:111548","journal-title":"J Comput Phys"},{"issue":"1","key":"3002_CR11","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1007\/s42967-020-00118-x","volume":"4","author":"J Du","year":"2022","unstructured":"Du J, Chung E, Yang Y (2022) Maximum-principle-preserving local discontinuous Galerkin methods for Allen\u2013Cahn equations. Commun Appl Math Comput 4(1):353\u2013379","journal-title":"Commun Appl Math Comput"},{"key":"3002_CR12","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/j.physd.2016.04.006","volume":"333","author":"GA El","year":"2016","unstructured":"El GA, Hoefer MA (2016) Dispersive shock waves and modulation theory. Physica D 333:11\u201365","journal-title":"Physica D"},{"issue":"12","key":"3002_CR13","doi-asserted-by":"publisher","first-page":"1569","DOI":"10.1002\/cpa.10050","volume":"55","author":"T Grava","year":"2002","unstructured":"Grava T, Tian F-R (2002) The generation, propagation, and extinction of multiphases in the KdV zero-dispersion limit. Commun Pure Appl Math 55(12):1569\u20131639","journal-title":"Commun Pure Appl Math"},{"key":"3002_CR14","first-page":"590","volume":"65","author":"AV Gurevich","year":"1973","unstructured":"Gurevich AV, Pitaevskii LP (1973) Nonstationary structure of a collisionless shock wave. Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 65:590\u2013604","journal-title":"Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki"},{"issue":"2","key":"3002_CR15","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1016\/0167-2789(90)90138-F","volume":"43","author":"H Iwasaki","year":"1990","unstructured":"Iwasaki H, Toh S, Kawahara T (1990) Cylindrical quasi-solitons of the Zakharov\u2013Kuznetsov equation. Physica D 43(2):293\u2013303","journal-title":"Physica D"},{"issue":"1","key":"3002_CR16","doi-asserted-by":"publisher","first-page":"250","DOI":"10.4208\/cicp.240815.301215a","volume":"20","author":"O Karakashian","year":"2016","unstructured":"Karakashian O, Xing Y (2016) A posteriori error estimates for conservative local discontinuous Galerkin methods for the generalized Korteweg\u2013de Vries equation. Commun Comput Phys 20(1):250\u2013278","journal-title":"Commun Comput Phys"},{"key":"3002_CR17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/1.9781611970562","volume-title":"Hyperbolic systems of conservation laws and the mathematical theory of shock waves","author":"PD Lax","year":"1973","unstructured":"Lax PD (1973) Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Society for Industrial and Applied Mathematics, Philadelphia, pp 1\u201348"},{"key":"3002_CR18","doi-asserted-by":"publisher","first-page":"2085","DOI":"10.1090\/mcom\/3550","volume":"89","author":"J Li","year":"2020","unstructured":"Li J, Zhang D, Meng X, Wu B (2020) Analysis of local discontinuous Galerkin methods with generalized numerical fluxes for linearized KdV equations. Math Comput 89:2085\u20132111","journal-title":"Math Comput"},{"key":"3002_CR19","doi-asserted-by":"publisher","first-page":"754","DOI":"10.1016\/j.jcp.2016.02.064","volume":"313","author":"D Luo","year":"2016","unstructured":"Luo D, Huang W, Qiu J (2016) A hybrid LDG-HWENO scheme for KdV-type equations. J Comput Phys 313:754\u2013774","journal-title":"J Comput Phys"},{"key":"3002_CR20","doi-asserted-by":"publisher","first-page":"370","DOI":"10.1016\/j.jcp.2017.03.024","volume":"339","author":"AK Meena","year":"2017","unstructured":"Meena AK, Kumar H, Chandrashekar P (2017) Positivity-preserving high-order discontinuous Galerkin schemes for Ten-Moment Gaussian closure equations. J Comput Phys 339:370\u2013395","journal-title":"J Comput Phys"},{"issue":"3","key":"3002_CR21","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1103\/PhysRevLett.46.184","volume":"46","author":"K Nozaki","year":"1981","unstructured":"Nozaki K (1981) Vortex solitons of drift waves and anomalous diffusion. Phys Rev Lett 46(3):184\u2013187","journal-title":"Phys Rev Lett"},{"issue":"293","key":"3002_CR22","first-page":"1145","volume":"84","author":"K Ohannes","year":"2015","unstructured":"Ohannes K, Makridakis C (2015) A posteriori error estimates for discontinuous Galerkin methods for the generalized Korteweg\u2013de Vries equation. Math Comput 84(293):1145\u20131167","journal-title":"Math Comput"},{"key":"3002_CR23","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1016\/j.jcp.2016.02.079","volume":"315","author":"T Qin","year":"2016","unstructured":"Qin T, Shu CW, Yang Y (2016) Bound-preserving discontinuous Galerkin methods for relativistic hydrodynamics. J Comput Phys 315:323\u2013347","journal-title":"J Comput Phys"},{"key":"3002_CR24","doi-asserted-by":"crossref","unstructured":"Shu CW (2018) Bound-preserving high-order schemes for hyperbolic equations: survey and recent developments","DOI":"10.1007\/978-3-319-91548-7_44"},{"issue":"9","key":"3002_CR25","doi-asserted-by":"publisher","first-page":"5472","DOI":"10.1103\/PhysRevA.40.5472","volume":"40","author":"S Toh","year":"1989","unstructured":"Toh S, Iwasaki H, Kawahara T (1989) Two-dimensionally localized pulses of a nonlinear equation with dissipation and dispersion. Phys Rev A 40(9):5472","journal-title":"Phys Rev A"},{"issue":"4","key":"3002_CR26","doi-asserted-by":"publisher","first-page":"1263","DOI":"10.3934\/era.2022066","volume":"30","author":"L Wei","year":"2022","unstructured":"Wei L, Wei X, Tang B (2022) Numerical analysis of variable-order fractional KdV\u2013Burgers\u2013Kuramoto equation. Electron Res Arch 30(4):1263\u20131281","journal-title":"Electron Res Arch"},{"issue":"4","key":"3002_CR27","doi-asserted-by":"publisher","first-page":"956","DOI":"10.1137\/0114075","volume":"14","author":"GB Whitham","year":"1965","unstructured":"Whitham GB (1965) Nonlinear dispersive waves. SIAM J Appl Math 14(4):956\u2013958","journal-title":"SIAM J Appl Math"},{"key":"3002_CR28","volume-title":"Linear and nonlinear waves","author":"GB Whitham","year":"1974","unstructured":"Whitham GB (1974) Linear and nonlinear waves, vol 635. John Wiley and Sons, Hoboken"},{"issue":"12","key":"3002_CR29","doi-asserted-by":"publisher","first-page":"1476","DOI":"10.1016\/j.advwatres.2010.08.005","volume":"33","author":"Y Xing","year":"2010","unstructured":"Xing Y, Zhang X, Shu CW (2010) Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv Water Resour 33(12):1476\u20131493","journal-title":"Adv Water Resour"},{"issue":"2","key":"3002_CR30","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1137\/140965326","volume":"37","author":"T Xiong","year":"2015","unstructured":"Xiong T, Qiu JM, Xu Z (2015) High order maximum-principle-preserving discontinuous Galerkin method for convection\u2013diffusion equations. SIAM J Sci Comput 37(2):583\u2013608","journal-title":"SIAM J Sci Comput"},{"key":"3002_CR31","doi-asserted-by":"publisher","first-page":"2213","DOI":"10.1090\/S0025-5718-2013-02788-3","volume":"83","author":"Z Xu","year":"2014","unstructured":"Xu Z (2014) Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem. Math Comput 83:2213\u20132238","journal-title":"Math Comput"},{"issue":"1\u20132","key":"3002_CR32","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/j.physd.2005.06.007","volume":"208","author":"Y Xu","year":"2005","unstructured":"Xu Y, Shu CW (2005) Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations. Physica D 208(1\u20132):21\u201358","journal-title":"Physica D"},{"issue":"37","key":"3002_CR33","doi-asserted-by":"publisher","first-page":"3805","DOI":"10.1016\/j.cma.2006.10.043","volume":"196","author":"Y Xu","year":"2007","unstructured":"Xu Y, Shu CW (2007) Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection\u2013diffusion and KdV equations. Comput Methods Appl Mech Eng 196(37):3805\u20133822","journal-title":"Comput Methods Appl Mech Eng"},{"key":"3002_CR34","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1016\/j.jcp.2019.03.046","volume":"390","author":"Z Xu","year":"2019","unstructured":"Xu Z, Yang Y, Guo H (2019) High-order bound-preserving discontinuous Galerkin methods for wormhole propagation on triangular meshes. J Comput Phys 390:323\u2013341","journal-title":"J Comput Phys"},{"key":"3002_CR35","doi-asserted-by":"crossref","unstructured":"Yan J, Shu CW (2002a) A local discontinuous Galerkin method for KdV type equations. SIAM J Numer Anal 40(2):769\u2013791","DOI":"10.1137\/S0036142901390378"},{"key":"3002_CR36","unstructured":"Yan J, Shu CW (2002b) Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J Sci Comput 17(1\u20134):27\u201347"},{"issue":"1","key":"3002_CR37","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1007\/s10915-022-02061-w","volume":"94","author":"X Yin","year":"2022","unstructured":"Yin X, Cao W (2022) A class of efficient hamiltonian conservative spectral methods for Korteweg\u2013de Vries equations. J Sci Comput 94(1):10","journal-title":"J Sci Comput"},{"issue":"66","key":"3002_CR38","first-page":"594","volume":"29","author":"VE Zakharov","year":"1974","unstructured":"Zakharov VE, Kuznetsov EA (1974) Three-dimensional solitons. Zhurnal Eksperimentalnoi I Teroreticheskoi Fiziki 29(66):594\u2013597","journal-title":"Zhurnal Eksperimentalnoi I Teroreticheskoi Fiziki"},{"key":"3002_CR39","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1016\/j.jcp.2016.10.002","volume":"328","author":"X Zhang","year":"2017","unstructured":"Zhang X (2017) On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier\u2013Stokes equations. J Comput Phys 328:301\u2013343","journal-title":"J Comput Phys"},{"key":"3002_CR40","doi-asserted-by":"crossref","unstructured":"Zhang X, Shu CW (2010a) On maximum-principle-satisfying high order schemes for scalar conservation laws. J Comput Phys 229(9):3091\u20133120","DOI":"10.1016\/j.jcp.2009.12.030"},{"key":"3002_CR41","doi-asserted-by":"crossref","unstructured":"Zhang X, Shu CW (2010b) On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J Comput Phys 229(23):8918\u20138934","DOI":"10.1016\/j.jcp.2010.08.016"},{"issue":"2","key":"3002_CR42","doi-asserted-by":"publisher","first-page":"532","DOI":"10.4208\/cicp.OA-2017-0204","volume":"25","author":"Q Zhang","year":"2019","unstructured":"Zhang Q, Xia Y (2019) Conservative and dissipative local discontinuous Galerkin methods for Korteweg\u2013de Vries type equations. Commun Comput Phys 25(2):532\u2013563","journal-title":"Commun Comput Phys"},{"key":"3002_CR43","doi-asserted-by":"crossref","unstructured":"Zhang F, Liu T, Liu M (2021a) A high-order maximum-principle-satisfying discontinuous Galerkin method for the level set problem. J Sci Comput 87(2):45","DOI":"10.1007\/s10915-021-01459-2"},{"key":"3002_CR44","doi-asserted-by":"crossref","unstructured":"Zhang C, Xu Y, Xia Y (2021b) Local discontinuous Galerkin methods to a dispersive system of KdV-type equations. J Sci Comput 86(1):1\u201343","DOI":"10.1007\/s10915-020-01370-2"},{"key":"3002_CR45","doi-asserted-by":"publisher","first-page":"198","DOI":"10.1016\/j.jcp.2015.12.039","volume":"308","author":"C Zheng","year":"2016","unstructured":"Zheng C, Huang H, Yan J (2016) Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangle mesh. J Comput Phys 308:198\u2013217","journal-title":"J Comput Phys"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-03002-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-024-03002-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-03002-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,4]],"date-time":"2025-03-04T05:16:51Z","timestamp":1741065411000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-024-03002-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,23]]},"references-count":45,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,2]]}},"alternative-id":["3002"],"URL":"https:\/\/doi.org\/10.1007\/s40314-024-03002-z","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2024,11,23]]},"assertion":[{"value":"16 April 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 August 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 October 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 November 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"There is no conflict of interest for the authors.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This article does not contain any studies with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical standard"}},{"value":"There is no individual participant included in the study.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Informed consent"}}],"article-number":"46"}}