{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T13:47:40Z","timestamp":1769521660444,"version":"3.49.0"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,2,19]],"date-time":"2020-02-19T00:00:00Z","timestamp":1582070400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,2,19]],"date-time":"2020-02-19T00:00:00Z","timestamp":1582070400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"National Key R&D Program of China","award":["2018YFA0703900"],"award-info":[{"award-number":["2018YFA0703900"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11631004"],"award-info":[{"award-number":["11631004"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-16-1-0103"],"award-info":[{"award-number":["W911NF-16-1-0103"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1719410"],"award-info":[{"award-number":["DMS-1719410"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Key R&D Program of China","award":["2018YFA0703900"],"award-info":[{"award-number":["2018YFA0703900"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11631004"],"award-info":[{"award-number":["11631004"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2020,3]]},"DOI":"10.1007\/s10915-020-01162-8","type":"journal-article","created":{"date-parts":[[2020,2,19]],"date-time":"2020-02-19T06:12:17Z","timestamp":1582092737000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["An Ultra-Weak Discontinuous Galerkin Method with Implicit\u2013Explicit Time-Marching for Generalized Stochastic KdV Equations"],"prefix":"10.1007","volume":"82","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9232-7144","authenticated-orcid":false,"given":"Yunzhang","family":"Li","sequence":"first","affiliation":[]},{"given":"Chi-Wang","family":"Shu","sequence":"additional","affiliation":[]},{"given":"Shanjian","family":"Tang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,2,19]]},"reference":[{"key":"1162_CR1","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/S0168-9274(97)00056-1","volume":"25","author":"UM Ascher","year":"1997","unstructured":"Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge\u2013Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151\u2013167 (1997)","journal-title":"Appl. Numer. Math."},{"key":"1162_CR2","doi-asserted-by":"publisher","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., Xing, Y.: Conservative, discontinuous Galerkin-methods for the generalized Korteweg\u2013de Vries equation. Math. Comput. 82, 1401\u20131432 (2013)","journal-title":"Math. Comput."},{"issue":"1","key":"1162_CR3","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1137\/S0036142995285873","volume":"35","author":"O Cessenat","year":"1998","unstructured":"Cessenat, O., Despres, B.: Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal. 35(1), 255\u2013299 (1998)","journal-title":"SIAM J. Numer. Anal."},{"key":"1162_CR4","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1088\/0741-3335\/28\/4\/005","volume":"28","author":"HY Chang","year":"1986","unstructured":"Chang, H.Y., Lien, C., Sukarto, S., Raychaudhury, S., Hill, J., Tsikis, E.K., Lonngren, K.E.: Propagation of ion-acoustic solitons in a non-quiescent plasma. Plasma Phys. Controlled Fusion 28, 675\u2013681 (1986)","journal-title":"Plasma Phys. Controlled Fusion"},{"issue":"262","key":"1162_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, C.-W.: A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Math. Comput. 77(262), 699\u2013730 (2008)","journal-title":"Math. Comput."},{"key":"1162_CR6","volume-title":"The Finite Element Method for Elliptic Problem","author":"P Ciarlet","year":"1975","unstructured":"Ciarlet, P.: The Finite Element Method for Elliptic Problem. North Holland, Amsterdam (1975)"},{"issue":"190","key":"1162_CR7","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."},{"key":"1162_CR8","doi-asserted-by":"publisher","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\u2013Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84, 90\u2013113 (1989)","journal-title":"J. Comput. Phys."},{"key":"1162_CR9","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1051\/m2an\/1991250303371","volume":"25","author":"B Cockburn","year":"1991","unstructured":"Cockburn, B., Shu, C.-W.: The Runge\u2013Kutta local projection P1-discontinuous Galerkin method for scalar conservation laws. Math. Model. Numer. Anal. 25, 337\u2013361 (1991)","journal-title":"Math. Model. Numer. Anal."},{"issue":"186","key":"1162_CR10","first-page":"411","volume":"52","author":"B Cockburn","year":"1989","unstructured":"Cockburn, B., Shu, C.-W.: TVB Runge\u2013Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52(186), 411\u2013435 (1989)","journal-title":"Math. Comput."},{"key":"1162_CR11","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1006\/jcph.1998.5892","volume":"141","author":"B Cockburn","year":"1998","unstructured":"Cockburn, B., Shu, C.-W.: The Runge\u2013Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199\u2013224 (1998)","journal-title":"J. Comput. Phys."},{"key":"1162_CR12","doi-asserted-by":"publisher","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, 2440\u20132463 (1998)","journal-title":"SIAM J. Numer. Anal."},{"key":"1162_CR13","first-page":"827","volume":"13","author":"T Dankel Jr","year":"2000","unstructured":"Dankel Jr., T.: On the stochastic Korteweg\u2013de Veries equation driven by white noise. Differ. Integral Equ. 13, 827\u2013836 (2000)","journal-title":"Differ. Integral Equ."},{"key":"1162_CR14","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.: Numerical simulation of the stochastic Korteweg\u2013de Vries equation. Physica D 134, 200\u2013226 (1999)","journal-title":"Physica D"},{"key":"1162_CR15","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1006\/jfan.1997.3184","volume":"154","author":"A de Bouard","year":"1998","unstructured":"de Bouard, A., Debussche, A.: On the stochastic Korteweg\u2013de Vries equation. J. Funct. Anal. 154, 215\u2013251 (1998)","journal-title":"J. Funct. Anal."},{"issue":"3","key":"1162_CR16","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1137\/S0036141003425301","volume":"36","author":"A de Bouard","year":"2004","unstructured":"de Bouard, A., Debussche, A., Tsutsumi, Y.: Periodic solutions of the Korteweg\u2013de Vries equation driven by white noise. SIAM J. Math. Anal. 36(3), 815\u2013855 (2004)","journal-title":"SIAM J. Math. Anal."},{"key":"1162_CR17","first-page":"97","volume":"310","author":"P Gao","year":"2017","unstructured":"Gao, P.: The stochastic Korteweg\u2013de Vries equation on a bounded domain. Appl. Math. Comput. 310, 97\u2013111 (2017)","journal-title":"Appl. Math. Comput."},{"key":"1162_CR18","doi-asserted-by":"publisher","first-page":"1063","DOI":"10.1088\/0305-4470\/23\/7\/014","volume":"23","author":"R Herman","year":"1990","unstructured":"Herman, R.: The stochastic, damped Korteweg\u2013de Vries equation. J. Phys. A Math. Gen. 23, 1063\u20131084 (1990)","journal-title":"J. Phys. A Math. Gen."},{"key":"1162_CR19","unstructured":"Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations, 3rd ed., Vol. 23, Applications in Mathematics, Stochastic Modeling and Applied Probability, Springer, Berlin (1999)"},{"key":"1162_CR20","doi-asserted-by":"publisher","first-page":"422","DOI":"10.1080\/14786449508620739","volume":"39","author":"DJ Korteweg","year":"1895","unstructured":"Korteweg, D.J., de Veries, G.: On the change of form of long waves advancing in rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39, 422\u2013443 (1895)","journal-title":"Philos. Mag."},{"issue":"1","key":"1162_CR21","doi-asserted-by":"publisher","first-page":"A54","DOI":"10.1137\/19M125710X","volume":"42","author":"Y Li","year":"2020","unstructured":"Li, Y., Shu, C.-W., Tang, S.: A discontinuous Galerkin method for stochastic conservation laws. SIAM J. Sci. Comput. 42(1), A54\u2013A86 (2020)","journal-title":"SIAM J. Sci. Comput."},{"key":"1162_CR22","unstructured":"Li, Y., Shu, C.-W., Tang, S.: A local discontinuous Galerkin method for nonlinear parabolic SPDEs, submitted"},{"issue":"2","key":"1162_CR23","doi-asserted-by":"publisher","first-page":"676","DOI":"10.1016\/j.jcp.2005.08.029","volume":"213","author":"G Lin","year":"2006","unstructured":"Lin, G., Grinberg, L., Karniadakis, G.E.: Numerical studies of the stochastic Korteweg\u2013de Vries equation. J. Comput. Phys. 213(2), 676\u2013703 (2006)","journal-title":"J. Comput. Phys."},{"key":"1162_CR24","doi-asserted-by":"publisher","DOI":"10.1533\/9780857099402","volume-title":"Stochastic Differential Equations and Applications","author":"X Mao","year":"2008","unstructured":"Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Horwood, Chichester (2008)","edition":"2"},{"key":"1162_CR25","doi-asserted-by":"crossref","unstructured":"Milstein, G.N.: Numerical Integration of Stochastic Differential Equations, Mathematics and its Applications, vol. 313. Kluwer Academic Publishers Group, Dordrecht, 1995 (Translated and revised from the 1988 Russian original) (1995)","DOI":"10.1007\/978-94-015-8455-5"},{"key":"1162_CR26","doi-asserted-by":"crossref","unstructured":"Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics, 1st ed., Scientific Computation series, Springer, Berlin (2004)","DOI":"10.1007\/978-3-662-10063-9_1"},{"key":"1162_CR27","volume-title":"Stochastic Integration and Differential Equations","author":"P Protter","year":"2004","unstructured":"Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, New York (2004)","edition":"2"},{"key":"1162_CR28","unstructured":"Reed, W.H., Hill, T.: Triangular mesh methods for the neutron transport equation, Technical report, Los Alamos Scientific Lab., N. Mex. (USA) (1973)"},{"key":"1162_CR29","volume-title":"Numerical Simulations of the Stochastic KdV Equation","author":"A Rose","year":"2006","unstructured":"Rose, A.: Numerical Simulations of the Stochastic KdV Equation. University of North Carolina, Wilmington (2006)"},{"key":"1162_CR30","doi-asserted-by":"publisher","first-page":"4166","DOI":"10.1103\/PhysRevE.58.4166","volume":"58","author":"M Scalerandi","year":"1998","unstructured":"Scalerandi, M., Romano, A., Condat, C.A.: Korteweg\u2013de Vries solitons under additive stochastic perturbations. Phys. Rev. E 58, 4166\u20134173 (1998)","journal-title":"Phys. Rev. E"},{"key":"1162_CR31","doi-asserted-by":"crossref","unstructured":"Shu, C.-W.: Discontinuous Galerkin methods for time-dependent convection dominated problems: Basics, recent developments and comparison with other methods, In: Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Springer, pp. 369-397 (2016)","DOI":"10.1007\/978-3-319-41640-3_12"},{"key":"1162_CR32","first-page":"237","volume":"272","author":"H-J Wang","year":"2016","unstructured":"Wang, H.-J., Shu, C.-W., Zhang, Q.: Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for nonlinear convection-diffusion problems. Appl. Math. Comput. 272, 237\u2013258 (2016)","journal-title":"Appl. Math. Comput."},{"key":"1162_CR33","doi-asserted-by":"publisher","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."},{"key":"1162_CR34","volume-title":"A Synergetic Approach to Problems of Nonlinear Dispersive Wave Propagation and Interaction, Nonlinear Partial Differential Equations","author":"NJ Zabusky","year":"1967","unstructured":"Zabusky, N.J.: A Synergetic Approach to Problems of Nonlinear Dispersive Wave Propagation and Interaction, Nonlinear Partial Differential Equations. Academic Press, New York (1967)"},{"key":"1162_CR35","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1137\/S0036142902404182","volume":"42","author":"Q Zhang","year":"2004","unstructured":"Zhang, Q., Shu, C.-W.: Error estimates to smooth solutions of Runge\u2013Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42, 641\u2013666 (2004)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01162-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-020-01162-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01162-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,15]],"date-time":"2022-10-15T23:19:01Z","timestamp":1665875941000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-020-01162-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,19]]},"references-count":35,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,3]]}},"alternative-id":["1162"],"URL":"https:\/\/doi.org\/10.1007\/s10915-020-01162-8","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,19]]},"assertion":[{"value":"24 October 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 January 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 February 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 February 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"61"}}