{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T09:00:27Z","timestamp":1772787627496,"version":"3.50.1"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,6,21]],"date-time":"2022-06-21T00:00:00Z","timestamp":1655769600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,6,21]],"date-time":"2022-06-21T00:00:00Z","timestamp":1655769600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2022,8]]},"DOI":"10.1007\/s10444-022-09944-4","type":"journal-article","created":{"date-parts":[[2022,6,21]],"date-time":"2022-06-21T05:03:53Z","timestamp":1655787833000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime"],"prefix":"10.1007","volume":"48","author":[{"given":"Teng","family":"Zhang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4878-070X","authenticated-orcid":false,"given":"Tingchun","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,6,21]]},"reference":[{"issue":"1","key":"9944_CR1","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/0022-1236(88)90036-5","volume":"79","author":"H Added","year":"1988","unstructured":"Added, H., Added, S.: Equations of Langmuir turbulence and nonlinear schr\u00f6dinger equation: smoothness and approximation. J. Funct. Anal. 79(1), 183\u2013210 (1988)","journal-title":"J. Funct. Anal."},{"issue":"2","key":"9944_CR2","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1137\/110830800","volume":"50","author":"W Bao","year":"2012","unstructured":"Bao, W., Cai, Y.: Uniform error estimates of finite difference methods for the nonlinear schr\u00f6dinger equation with wave operator. SIAM J. Numer. Anal. 50(2), 492\u2013521 (2012)","journal-title":"SIAM J. Numer. Anal."},{"issue":"281","key":"9944_CR3","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1090\/S0025-5718-2012-02617-2","volume":"82","author":"W Bao","year":"2013","unstructured":"Bao, W., Cai, Y.: Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation. Math. Comput. 82(281), 99\u2013128 (2013)","journal-title":"Math. Comput."},{"issue":"6","key":"9944_CR4","doi-asserted-by":"publisher","first-page":"A2903","DOI":"10.1137\/110855004","volume":"35","author":"W Bao","year":"2013","unstructured":"Bao, W., Dong, X., Zhao, X.: An exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system. SIAM J. Sci. Comput. 35(6), A2903\u2013A2927 (2013)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"9944_CR5","doi-asserted-by":"publisher","first-page":"977","DOI":"10.1137\/16M1078112","volume":"15","author":"W Bao","year":"2017","unstructured":"Bao, W., Su, C.: Uniform error bounds of a finite difference method for the Zakharov system in the subsonic limit regime via an asymptotic consistent formulation. Multiscale Model. Sim. 15(2), 977\u20131002 (2017)","journal-title":"Multiscale Model. Sim."},{"issue":"2","key":"9944_CR6","doi-asserted-by":"publisher","first-page":"A929","DOI":"10.1137\/17M1113333","volume":"40","author":"W Bao","year":"2018","unstructured":"Bao, W., Su, C.: A uniformly and optimally accurate method for the Zakharov system in the subsonic limit regime. SIAM. J. Sci. Comput. 40(2), A929\u2013A953 (2018)","journal-title":"SIAM. J. Sci. Comput."},{"issue":"3","key":"9944_CR7","doi-asserted-by":"publisher","first-page":"1057","DOI":"10.1137\/030600941","volume":"26","author":"W Bao","year":"2005","unstructured":"Bao, W., Sun, F.: Efficient and stable numerical methods for the generalized and vector Zakharov system. SIAM J. Sci. Comput. 26(3), 1057\u20131088 (2005)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"9944_CR8","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1016\/S0021-9991(03)00271-7","volume":"190","author":"W Bao","year":"2003","unstructured":"Bao, W., Sun, F., Wei, G.W.: Numerical methods for the generalized Zakharov system. J. Comput. Phys. 190(1), 201\u2013228 (2003)","journal-title":"J. Comput. Phys."},{"key":"9944_CR9","doi-asserted-by":"publisher","first-page":"108886","DOI":"10.1016\/j.jcp.2019.108886","volume":"398","author":"W Bao","year":"2019","unstructured":"Bao, W., Zhao, X.: Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime. J. Comput. Phys. 398, 108886 (2019)","journal-title":"J. Comput. Phys."},{"issue":"11","key":"9944_CR10","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1155\/S1073792896000359","volume":"1996","author":"J Bourgain","year":"1996","unstructured":"Bourgain, J., Colliander, J.: On wellposedness of the Zakharov system. Int. Math. Res. Notices 1996(11), 515\u2013546 (1996)","journal-title":"Int. Math. Res. Notices"},{"issue":"311","key":"9944_CR11","doi-asserted-by":"publisher","first-page":"1191","DOI":"10.1090\/mcom\/3269","volume":"87","author":"Y Cai","year":"2018","unstructured":"Cai, Y., Yuan, Y.: Uniform error estimates of the conservative finite difference method for the Zakharov system in the subsonic limit regime. Math. Comput. 87(311), 1191\u20131225 (2018)","journal-title":"Math. Comput."},{"issue":"210","key":"9944_CR12","doi-asserted-by":"publisher","first-page":"537","DOI":"10.1090\/S0025-5718-1995-1284664-5","volume":"64","author":"QS Chang","year":"1995","unstructured":"Chang, Q.S., Guo, B.L., Jiang, H.: Finite difference method for generalized Zakharov equations. Math. Comput. 64(210), 537\u2013553 (1995)","journal-title":"Math. Comput."},{"issue":"2","key":"9944_CR13","doi-asserted-by":"publisher","first-page":"384","DOI":"10.1006\/jfan.1997.3148","volume":"151","author":"J Ginibre","year":"1997","unstructured":"Ginibre, J., Tsutsumi, Y., Velo, G.: On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151(2), 384\u2013436 (1997)","journal-title":"J. Funct. Anal."},{"issue":"1","key":"9944_CR14","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1007\/BF02099792","volume":"160","author":"L Glangetas","year":"1994","unstructured":"Glangetas, L., Merle, F., et al.: Existence of self-similar blow-up solutions for Zakharov equation in dimension two. i. Commun. Math. phys. 160 (1), 173\u2013215 (1994)","journal-title":"Commun. Math. phys."},{"issue":"2","key":"9944_CR15","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1016\/0021-9991(92)90243-R","volume":"100","author":"RT Glassey","year":"1992","unstructured":"Glassey, R.T.: Approximate solutions to the Zakharov equations via finite differences. J. Comput. Phys. 100(2), 377\u2013383 (1992)","journal-title":"J. Comput. Phys."},{"issue":"197","key":"9944_CR16","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1090\/S0025-5718-1992-1106968-6","volume":"58","author":"RT Glassey","year":"1992","unstructured":"Glassey, R.T.: Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension. Math. Comput. 58(197), 83\u2013102 (1992)","journal-title":"Math. Comput."},{"issue":"1","key":"9944_CR17","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1016\/j.jcp.2004.06.001","volume":"201","author":"S Jin","year":"2004","unstructured":"Jin, S., Markowich, P.A., Zheng, C.: Numerical simulation of a generalized Zakharov system. J. Comput. Phys. 201(1), 376\u2013395 (2004)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"9944_CR18","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/0021-9991(92)90324-R","volume":"103","author":"SK Lele","year":"1992","unstructured":"Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16\u201342 (1992)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"9944_CR19","doi-asserted-by":"publisher","first-page":"602","DOI":"10.1016\/j.camwa.2018.04.040","volume":"76","author":"Y Ma","year":"2018","unstructured":"Ma, Y., Su, C.: A uniformly and optimally accurate multiscale time integrator method for the Klein-Gordon-Zakharov system in the subsonic limit regime. Comput. Math. Appl. 76(3), 602\u2013619 (2018)","journal-title":"Comput. Math. Appl."},{"issue":"3","key":"9944_CR20","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1007\/s00222-008-0110-5","volume":"172","author":"N Masmoudi","year":"2008","unstructured":"Masmoudi, N., Nakanishi, K.: Energy convergence for singular limits of Zakharov type systems. Invent. Math. 172(3), 535\u2013583 (2008)","journal-title":"Invent. Math."},{"issue":"2","key":"9944_CR21","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/BF02102415","volume":"175","author":"F Merle","year":"1996","unstructured":"Merle, F.: Blow-up results of viriel type for Zakharov equations. Commun. Math. Phys. 175(2), 433\u2013455 (1996)","journal-title":"Commun. Math. Phys."},{"issue":"4","key":"9944_CR22","first-page":"721","volume":"5","author":"T Ozawa","year":"1992","unstructured":"Ozawa, T., Tsutsumi, Y., Brezis, H.: The nonlinear schr\u00f6dinger limit and the initial layer of the Zakharov equations. Differ. Integral Equ. 5(4), 721\u2013745 (1992)","journal-title":"Differ. Integral Equ."},{"issue":"4","key":"9944_CR23","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1007\/BF01463396","volume":"106","author":"SH Schochet","year":"1986","unstructured":"Schochet, S.H., Weinstein, M.I.: The nonlinear schr\u00f6dinger limit of the Zakharov equations governing Langmuir turbulence. Comm. Math. Phys. 106(4), 569\u2013580 (1986)","journal-title":"Comm. Math. Phys."},{"key":"9944_CR24","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1016\/j.cam.2017.09.003","volume":"330","author":"C Su","year":"2018","unstructured":"Su, C.: Comparison of numerical methods for the Zakharov system in the subsonic limit regime. J. Comput. Appl. Math. 330, 441\u2013455 (2018)","journal-title":"J. Comput. Appl. Math."},{"issue":"7","key":"9944_CR25","doi-asserted-by":"publisher","first-page":"1063","DOI":"10.1016\/j.cpc.2008.12.028","volume":"180","author":"J Wang","year":"2009","unstructured":"Wang, J.: Multisymplectic numerical method for the Zakharov system. Comput. Phys. Comm. 180(7), 1063\u20131071 (2009)","journal-title":"Comput. Phys. Comm."},{"issue":"3","key":"9944_CR26","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1007\/s10092-018-0277-0","volume":"55","author":"T Wang","year":"2018","unstructured":"Wang, T., Zhao, X.: Unconditional $ {L}^{\\infty }$-convergence of two compact conservative finite difference schemes for the nonlinear schr\u00f6dinger equation in multi-dimensions. Calcolo 55(3), 34 (2018)","journal-title":"Calcolo"},{"issue":"4","key":"9944_CR27","doi-asserted-by":"publisher","first-page":"1238","DOI":"10.1016\/j.jcp.2009.10.029","volume":"229","author":"Y Xia","year":"2010","unstructured":"Xia, Y., Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for the generalized Zakharov system. J. Comput. Phys. 229(4), 1238\u20131259 (2010)","journal-title":"J. Comput. Phys."},{"issue":"5","key":"9944_CR28","first-page":"908","volume":"35","author":"VE Zakharov","year":"1972","unstructured":"Zakharov, V.E.: Collapse of Langmuir waves. Sov. Phys. JETP 35(5), 908\u2013914 (1972)","journal-title":"Sov. Phys. JETP"},{"issue":"3","key":"9944_CR29","doi-asserted-by":"publisher","first-page":"2089","DOI":"10.1002\/num.22664","volume":"37","author":"T Zhang","year":"2021","unstructured":"Zhang, T., Wang, T.: Optimal error estimates of fourth\u2013order compact finite difference methods for the nonlinear Klein\u2013Gordon equation in the nonrelativistic regime. Numer. Methods Partial Differ. Equ. 37(3), 2089\u20132108 (2021)","journal-title":"Numer. Methods Partial Differ. Equ."},{"issue":"2","key":"9944_CR30","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1080\/00207160.2017.1284319","volume":"95","author":"X Zhou","year":"2018","unstructured":"Zhou, X., Zhang, L.: A conservative compact difference scheme for the Zakharov equations in one space dimension. Int. J. Comput. Math. 95(2), 279\u2013302 (2018)","journal-title":"Int. J. Comput. Math."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-022-09944-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-022-09944-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-022-09944-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,9]],"date-time":"2022-08-09T12:28:05Z","timestamp":1660048085000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-022-09944-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,21]]},"references-count":30,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,8]]}},"alternative-id":["9944"],"URL":"https:\/\/doi.org\/10.1007\/s10444-022-09944-4","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,21]]},"assertion":[{"value":"29 November 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 March 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 June 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}],"article-number":"40"}}