{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T04:09:13Z","timestamp":1778558953094,"version":"3.51.4"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2019,6,24]],"date-time":"2019-06-24T00:00:00Z","timestamp":1561334400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,6,24]],"date-time":"2019-06-24T00:00:00Z","timestamp":1561334400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2020,5]]},"DOI":"10.1007\/s11075-019-00761-3","type":"journal-article","created":{"date-parts":[[2019,6,24]],"date-time":"2019-06-24T22:55:06Z","timestamp":1561416906000},"page":"407-425","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Conservative Local Discontinuous Galerkin method for the fractional Klein-Gordon-Schr\u00f6dinger system with generalized Yukawa interaction"],"prefix":"10.1007","volume":"84","author":[{"given":"P.","family":"Castillo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.","family":"G\u00f3mez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,6,24]]},"reference":[{"key":"761_CR1","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1016\/j.cnsns.2017.06.018","volume":"54","author":"T Aboelenen","year":"2018","unstructured":"Aboelenen, T.: A high-order nodal Discontinuous Galerkin method for nonlinear fractional Schr\u00f6dinger type equations. Commun. Nonlinear Sci. Numer. Simul. 54, 428\u2013452 (2018)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"761_CR2","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.na.2017.01.006","volume":"155","author":"A Ardila","year":"2017","unstructured":"Ardila, A.: Existence and stability of standing waves for nonlinear fractional Schr\u00f6dinger equation with logarithmic nonlinearity. Nonlinear Anal. 155, 52\u201364 (2017)","journal-title":"Nonlinear Anal."},{"issue":"5","key":"761_CR3","doi-asserted-by":"crossref","first-page":"1749","DOI":"10.1137\/S0036142901384162","volume":"39","author":"D Arnold","year":"2002","unstructured":"Arnold, D., Brezzi, F., Cockburn, B., Marini, L.: Unified analysis of Discontinuous Galerkin methods for elliptic problems. SIAM J. Num. Anal. 39(5), 1749\u20131779 (2002)","journal-title":"SIAM J. Num. Anal."},{"key":"761_CR4","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.cma.2011.10.012","volume":"209-212","author":"M Baccouch","year":"2012","unstructured":"Baccouch, M.: A local discontinuous GAlerkin method for the second-order wave equation. Comput. Methods Appl. Mech. Engrg. 209-212, 129\u2013143 (2012)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"issue":"2","key":"761_CR5","doi-asserted-by":"crossref","first-page":"1863","DOI":"10.1016\/j.jcp.2007.02.018","volume":"225","author":"W Bao","year":"2007","unstructured":"Bao, W., Yang, L.: Efficient and accurate numerical methods for the Klein-Gordon-Schr\u00f6dinger equations. J. Comput. Phys. 225(2), 1863\u20131893 (2007)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"761_CR6","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1137\/S1064827501388339","volume":"24","author":"P Castillo","year":"2002","unstructured":"Castillo, P.: Performance of discontinuous Galerkin methods for elliptic PDEs. SIAM J. Sci. Comput. 24(2), 524\u2013547 (2002)","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"761_CR7","doi-asserted-by":"crossref","first-page":"1676","DOI":"10.1137\/S0036142900371003","volume":"38","author":"P Castillo","year":"2000","unstructured":"Castillo, P., Cockburn, B., Perugia, I., Sch\u00f6tzau, D.: An a priori error analysis of the Local Discontinuous Galerkin method for elliptic problems. SIAM J. Num. Anal. 38(5), 1676\u20131706 (2000)","journal-title":"SIAM J. Num. Anal."},{"issue":"3","key":"761_CR8","doi-asserted-by":"crossref","first-page":"1444","DOI":"10.1007\/s10915-018-0708-8","volume":"77","author":"P Castillo","year":"2018","unstructured":"Castillo, P., G\u00f3mez, S.: On the conservation of fractional nonlinear Schr\u00f6dinger\u2019s invariants by the Local Discontinuous Galerkin method. J. Sci. Comput. 77(3), 1444\u20131467 (2018)","journal-title":"J. Sci. Comput."},{"issue":"9","key":"761_CR9","doi-asserted-by":"crossref","first-page":"2279","DOI":"10.1016\/j.mcm.2011.07.032","volume":"57","author":"P Castillo","year":"2013","unstructured":"Castillo, P., Sequeira, F.: Computational aspects of the Local Discontinuous Galerkin method on unstructured grids in three dimensions. Math. Comput. Model. 57(9), 2279\u20132288 (2013)","journal-title":"Math. Comput. Model."},{"key":"761_CR10","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/s10915-007-9130-3","volume":"32","author":"B Cockburn","year":"2007","unstructured":"Cockburn, B., Dong, B.: An analysis of the minimal dissipation Local Discontinuous Galerkin method for convection diffusion problems. SIAM J. Sci. Comput. 32, 233\u2013262 (2007)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"761_CR11","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1137\/S0036142900371544","volume":"39","author":"B Cockburn","year":"2001","unstructured":"Cockburn, B., Kanschat, G., Perugia, I., Sch\u00f6tzau, D.: Superconvergence of the Local Discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Num. Anal. 39(1), 264\u2013285 (2001)","journal-title":"SIAM J. Num. Anal."},{"issue":"6","key":"761_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. Num. Anal. 35(6), 2440\u20132463 (1998)","journal-title":"SIAM J. Num. Anal."},{"issue":"18","key":"761_CR13","doi-asserted-by":"crossref","first-page":"5207","DOI":"10.1002\/mma.3449","volume":"38","author":"P D\u2019Avenia","year":"2015","unstructured":"D\u2019Avenia, P., Squassina, M., Zenari, M.: Fractional logarithmic Schr\u00f6dinger equations. Math. Meth. Appl. Sci. 38(18), 5207\u20135216 (2015)","journal-title":"Math. Meth. Appl. Sci."},{"issue":"6","key":"761_CR14","doi-asserted-by":"crossref","first-page":"2340","DOI":"10.1016\/j.apm.2011.08.030","volume":"36","author":"M Dehghan","year":"2012","unstructured":"Dehghan, M., Taleei, A.: Numerical solution of the Yukawa-coupled Klein-Gordon-Schr\u00f6dinger equations via a Chebyshev pseudospectral multidomain method. Appl. Math. Model. 36(6), 2340\u20132349 (2012)","journal-title":"Appl. Math. Model."},{"issue":"2","key":"761_CR15","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/0021-9991(81)90052-8","volume":"44","author":"M Delfour","year":"1981","unstructured":"Delfour, M., Fortin, M., Payr\u00e9, G.: Finite-difference solutions of a non-linear Schr\u00f6dinger equation. J. Comput. Phys. 44(2), 277\u2013288 (1981)","journal-title":"J. Comput. Phys."},{"issue":"6","key":"761_CR16","doi-asserted-by":"crossref","first-page":"1237","DOI":"10.1017\/S0308210511000746","volume":"142","author":"P Felmer","year":"2012","unstructured":"Felmer, P., Quass, A., Tan, J.: Positive solutions of the nonlinear Schr\u00f6xdinger equation with the fractional Laplacian. Proc. R. Soc. Edinburgh Sect. A: Math. 142 (6), 1237\u20131262 (2012)","journal-title":"Proc. R. Soc. Edinburgh Sect. A: Math."},{"key":"761_CR17","first-page":"51","volume":"69","author":"I Fukuda","year":"1975","unstructured":"Fukuda, I., Tsutsumi, M.: On coupled Klein-Gordon-Schr\u00f6dinger equations, I. Bull. Sci. Engrg. Lab. Waseda Univ. 69, 51\u201362 (1975)","journal-title":"Bull. Sci. Engrg. Lab. Waseda Univ."},{"key":"761_CR18","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/0022-247X(78)90239-1","volume":"66","author":"I Fukuda","year":"1978","unstructured":"Fukuda, I., Tsutsumi, M.: On coupled Klein-Gordon-Schr\u00f6dinger equations, II. J. Math. Anal. and Appl. 66, 358\u2013378 (1978)","journal-title":"J. Math. Anal. and Appl."},{"issue":"1","key":"761_CR19","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1016\/j.amc.2008.07.003","volume":"204","author":"B Guo","year":"2008","unstructured":"Guo, B., Han, Y., Xin, J.: Existence of the global smooth solution to the period boundary value problem of fractional nonlinear S,chr\u00f6dinger equation. App Math. Comp. 204(1), 468\u2013477 (2008)","journal-title":"App Math. Comp."},{"issue":"2","key":"761_CR20","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1007\/s10915-014-9977-z","volume":"65","author":"L Guo","year":"2015","unstructured":"Guo, L., discontinuous, Y. X. u.: Energy conserving local Galerkin methods for the nonlinear Schr\u00f6dinger equation with wave operator. J. Sci. Comput. 65(2), 622\u2013647 (2015)","journal-title":"J. Sci. Comput."},{"issue":"8","key":"761_CR21","doi-asserted-by":"crossref","first-page":"082104","DOI":"10.1063\/1.2235026","volume":"47","author":"X Guo","year":"2006","unstructured":"Guo, X., Xu, M.: Some physical applications of fractional Schr\u00f6dinger equation. J. Math. Phys. 47(8), 082104 (2006)","journal-title":"J. Math. Phys."},{"key":"761_CR22","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.apnum.2018.01.004","volume":"127","author":"J Hong","year":"2018","unstructured":"Hong, J., Ji, L., Liu, Z.: Optimal error estimate of conservative local discontinuous Galerkin method for nonlinear Schr\u00f6dinger equation. Appl. Numer Math. 127, 164\u2013178 (2018)","journal-title":"Appl. Numer Math."},{"issue":"7","key":"761_CR23","doi-asserted-by":"crossref","first-page":"1345","DOI":"10.1007\/s11425-016-5133-6","volume":"59","author":"C Huang","year":"2016","unstructured":"Huang, C., Guo, G., Huang, D., Li, Q.: Global well-posedness of the fractional Klein-Gordon-Schr\u00f6dinger system with rough initial data. Sci. China Math. 59(7), 1345\u20131366 (2016)","journal-title":"Sci. China Math."},{"issue":"8","key":"761_CR24","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1016\/j.cpc.2010.04.003","volume":"181","author":"L Kong","year":"2010","unstructured":"Kong, L., Zhang, J., Cao, Y., Duan, Y., Huang, H.: Semi-explicit symplectic partitioned Runge-Kutta Fourier pseudo-spectral scheme for Klein-Gordon-Schr\u00f6dinger equations. Comput. Phys. Commun. 181(8), 1369\u20131377 (2010)","journal-title":"Comput. Phys. Commun."},{"issue":"3","key":"761_CR25","doi-asserted-by":"crossref","first-page":"3135","DOI":"10.1103\/PhysRevE.62.3135","volume":"62","author":"N Laskin","year":"2000","unstructured":"Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62(3), 3135 (2000)","journal-title":"Phys. Rev. E"},{"issue":"4","key":"761_CR26","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/S0375-9601(00)00201-2","volume":"268","author":"N Laskin","year":"2000","unstructured":"Laskin, N.: Fractional quantum mechanics and L\u00e9vy path integrals. Phys. Lett. A 268(4), 298\u2013305 (2000)","journal-title":"Phys. Lett. A"},{"issue":"5","key":"761_CR27","doi-asserted-by":"crossref","first-page":"056108","DOI":"10.1103\/PhysRevE.66.056108","volume":"66","author":"N Laskin","year":"2002","unstructured":"Laskin, N.: Fractional Schr\u00f6dinger equation. Phys. Rev. E 66(5), 056108 (2002)","journal-title":"Phys. Rev. E"},{"key":"761_CR28","first-page":"475","volume":"238","author":"H Liang","year":"2014","unstructured":"Liang, H.: Linearly implicit conservative schemes for long-term numerical simulation of Klein-Gordon-Schr\u00f6dinger equations. Appl. Math. Comput. 238, 475\u2013484 (2014)","journal-title":"Appl. Math. Comput."},{"key":"761_CR29","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.na.2013.07.003","volume":"92","author":"X Pan","year":"2013","unstructured":"Pan, X., Zhang, L.: High-order linear compact conservative method for the nonlinear Schr\u00f6dinger equation coupled with the nonlinear Klein-Gordon equation. Nonlinear Anal. Theory Methods Appl. 92, 108\u2013118 (2013)","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"issue":"167","key":"761_CR30","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1090\/S0025-5718-1984-0744922-X","volume":"43","author":"JM Sanz-Serna","year":"1984","unstructured":"Sanz-Serna, J.M.: Methods for the numerical solution of the nonlinear Schr\u00f6dinger equation. Math. Comp. 43(167), 21\u201327 (1984)","journal-title":"Math. Comp."},{"issue":"1","key":"761_CR31","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1093\/imanum\/6.1.25","volume":"6","author":"JM Sanz-Serna","year":"1986","unstructured":"Sanz-Serna, J.M., Verwer, J.G.: Conservative and nonconservative schemes for the solution of the nonlinear Schr\u00f6dinger equation. IMA J. Numer. Anal. 6(1), 25\u201342 (1986)","journal-title":"IMA J. Numer. Anal."},{"issue":"2","key":"761_CR32","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/0021-9991(78)90038-4","volume":"28","author":"W Strauss","year":"1978","unstructured":"Strauss, W., Vazquez, L.: Numerical solution of a nonlinear Klein-Gordon equation. J. Comput. Phys. 28(2), 271\u2013278 (1978)","journal-title":"J. Comput. Phys."},{"key":"761_CR33","first-page":"691","volume":"320","author":"J Wang","year":"2018","unstructured":"Wang, J., Xiao, A.: An efficient conservative difference scheme for fractional Klein-Gordon-Schr\u00f6dinger equations. Appl. Math. Comput. 320, 691\u2013709 (2018)","journal-title":"Appl. Math. Comput."},{"issue":"4","key":"761_CR34","doi-asserted-by":"crossref","first-page":"715","DOI":"10.4208\/eajam.220418.300618","volume":"8","author":"J Wang","year":"2018","unstructured":"Wang, J., Xiao, A., Wang, C.: A conservative difference scheme for space fractional Klein-Gordon-Schr\u00f6dinger equations with a high-degree Yukawa interation. East Asian J. Appl. Math. 8(4), 715\u2013745 (2018)","journal-title":"East Asian J. Appl. Math."},{"issue":"1","key":"761_CR35","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.jmaa.2013.10.038","volume":"412","author":"T Wang","year":"2014","unstructured":"Wang, T.: Optimal point-wise error estimate of a compact difference scheme for the Klein-Gordon-Schr\u00f6dinger equation. J. Math. Anal. Appl. 412(1), 155\u2013167 (2014)","journal-title":"J. Math. Anal. Appl."},{"key":"761_CR36","first-page":"239","volume":"10","author":"P Xanthopoulos","year":"2008","unstructured":"Xanthopoulos, P., Zouraris, G.: A linearly explicit finite different method for a Klein-Gordon-Schr\u00f6dinger system modeling electron-ion plasma waves. Discrete Contin. Dyn. Syst. Ser. B 10, 239\u2013263 (2008)","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"issue":"4","key":"761_CR37","doi-asserted-by":"crossref","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."},{"key":"761_CR38","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/0377-0427(88)90265-8","volume":"21","author":"X Xiang","year":"1988","unstructured":"Xiang, X.: Spectral method for solving the system of equations of Klein-Gordon-Schr\u00f6dinger field. J. Comput. Appl. Math. 21, 161\u2013171 (1988)","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"761_CR39","doi-asserted-by":"crossref","first-page":"967","DOI":"10.3934\/ipi.2013.7.967","volume":"7","author":"Y Xing","year":"2013","unstructured":"Xing, Y., Chou, C.S., Shu, C.W.: Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Probl. Imaging 7(3), 967\u2013986 (2013)","journal-title":"Inverse Probl. Imaging"},{"issue":"1","key":"761_CR40","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.jcp.2004.11.001","volume":"205","author":"Y Xu","year":"2005","unstructured":"Xu, Y., Shu, C.W.: Local Discontinuous Galerkin methods for nonlinear Schr\u00f6dinger equations. J. Comput. Phys. 205(1), 72\u201397 (2005)","journal-title":"J. Comput. Phys."},{"key":"761_CR41","unstructured":"Zhang, H., Hu, Q.: Existence of the global solution for fractional logarithmic Schr\u00f6dinger equation Comp. & Math with Applic. (2017)"},{"issue":"1","key":"761_CR42","first-page":"343","volume":"163","author":"L Zhang","year":"2005","unstructured":"Zhang, L.: Convergence of a conservative difference scheme for a class of Klein-Gordon-Schr\u00f6dinger equations in one space dimension. Appl. Math. Comput. 163(1), 343\u2013355 (2005)","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"761_CR43","doi-asserted-by":"crossref","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. of Comp. Math. 95(2), 279\u2013302 (2018)","journal-title":"Int J. of Comp. Math."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00761-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-019-00761-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00761-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,6,22]],"date-time":"2020-06-22T23:33:31Z","timestamp":1592868811000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-019-00761-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,24]]},"references-count":43,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,5]]}},"alternative-id":["761"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00761-3","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,24]]},"assertion":[{"value":"18 January 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 June 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 June 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}