{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,28]],"date-time":"2026-10-28T15:04:19Z","timestamp":1793199859210,"version":"4.5.0"},"reference-count":42,"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"}],"funder":[{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["JZ2021HGQA0246"],"award-info":[{"award-number":["JZ2021HGQA0246"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2022,8]]},"DOI":"10.1007\/s10444-022-09962-2","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":11,"title":["High-order conservative energy quadratization schemes for the Klein-Gordon-Schr\u00f6dinger equation"],"prefix":"10.1007","volume":"48","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6309-6765","authenticated-orcid":false,"given":"Xin","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luming","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,6,21]]},"reference":[{"key":"9962_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0370-1573(78)90074-1","volume":"35","author":"VG Makhankov","year":"1978","unstructured":"Makhankov, V.G.: Dynamics of classical solitons (in non-integrable systems). Phys. Rep. 35, 1\u2013128 (1978)","journal-title":"Phys. Rep."},{"key":"9962_CR2","doi-asserted-by":"publisher","first-page":"833","DOI":"10.1007\/s00211-016-0818-x","volume":"135","author":"W Bao","year":"2017","unstructured":"Bao, W., Zhao, X.: A uniformly accurate (UA) multiscale time integrator fourier pseoduspectral method for the Klein-Gordon-Schr\u00f6Dinger equations in the nonrelativistic limit regime. Numer. Math. 135, 833\u2013873 (2017)","journal-title":"Numer. Math."},{"key":"9962_CR3","doi-asserted-by":"publisher","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. Appl. 66, 358\u2013378 (1978)","journal-title":"J. Math. Anal. Appl."},{"key":"9962_CR4","first-page":"897","volume":"25","author":"B Guo","year":"1982","unstructured":"Guo, B.: Global solution for some problem of a class of equations in interaction of complex Schr\u00f6Dinger field and real Klein-Gordon field. Sci. China A 25, 897\u2013910 (1982)","journal-title":"Sci. China A"},{"key":"9962_CR5","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1016\/0362-546X(95)00017-P","volume":"27","author":"M Ohta","year":"1996","unstructured":"Ohta, M.: Stability of stationary states for the coupled Klein-Gordon-Schr\u00f6dinger equations. Nonlinear Anal. 27, 455\u2013461 (1996)","journal-title":"Nonlinear Anal."},{"key":"9962_CR6","first-page":"1444","volume":"38","author":"B Guo","year":"1995","unstructured":"Guo, B., Miao, C.: Global existence and asymptotic behavior of solutions for the coupled Klein-Gordon-Schr\u00f6dinger equations. Sci. China A 38, 1444\u20131456 (1995)","journal-title":"Sci. China A"},{"issue":"5","key":"9962_CR7","doi-asserted-by":"publisher","first-page":"1742","DOI":"10.1137\/100790586","volume":"8","author":"W Bao","year":"2010","unstructured":"Bao, W., Dong, X., Wang, S.: Singular limits of Klein-Gordon-Schr\u00f6dinger equations. Multiscale Model. Simul. 8(5), 1742\u20131769 (2010)","journal-title":"Multiscale Model. Simul."},{"key":"9962_CR8","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1016\/j.physleta.2003.07.026","volume":"318","author":"M Wang","year":"2003","unstructured":"Wang, M., Zhou, Y.: The periodic wave solutions for the Klein-Gordon-Schr\u00f6dinger equations. Phys. Lett. A 318, 84\u201392 (2003)","journal-title":"Phys. Lett. A"},{"key":"9962_CR9","doi-asserted-by":"publisher","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, 155\u2013167 (2014)","journal-title":"J. Math. Anal. Appl."},{"key":"9962_CR10","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1007\/s10444-017-9557-5","volume":"44","author":"T Wang","year":"2018","unstructured":"Wang, T., Zhao, X., Jiang, J.: Unconditional and optimal h2-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schr\u00f6dinger equation in high dimensions. Adv. Comput. Math. 44, 477\u2013503 (2018)","journal-title":"Adv. Comput. Math."},{"key":"9962_CR11","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1016\/j.cam.2019.02.018","volume":"358","author":"J Wang","year":"2019","unstructured":"Wang, J., Liang, D., Wang, Y.: Analysis of a conservative high-order compact finite difference scheme for the Klein-Gordon-Schr\u00f6dinger equation. J. Comput. Appl. Math. 358, 84\u201396 (2019)","journal-title":"J. Comput. Appl. Math."},{"key":"9962_CR12","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1016\/j.apnum.2019.02.004","volume":"142","author":"M Li","year":"2019","unstructured":"Li, M., Shi, D., Wang, J., Ming, W.: Unconditional superconvergence analysis of the conservative linearized Galerkin FEMs for nonlinear Klein-Gordon-Schr\u00f6dinger equation. Appl. Numer. Math. 142, 47\u201363 (2019)","journal-title":"Appl. Numer. Math."},{"key":"9962_CR13","first-page":"342","volume":"181","author":"L Kong","year":"2006","unstructured":"Kong, L., Liu, R., Xu, Z.: Numerical simulation of interaction between Schr\u00f6Dinger field and Klein-Gordon field by multisymplectic method. Appl. Math. Comput. 181, 342\u2013350 (2006)","journal-title":"Appl. Math. Comput."},{"key":"9962_CR14","doi-asserted-by":"publisher","first-page":"3517","DOI":"10.1016\/j.jcp.2009.02.006","volume":"228","author":"J Hong","year":"2009","unstructured":"Hong, J., Jiang, S., Li, C.: Explicit multi-symplectic methods for Klein-Gordon-Schr\u00f6Dinger equations. J. Comput. Phys. 228, 3517\u20133532 (2009)","journal-title":"J. Comput. Phys."},{"key":"9962_CR15","doi-asserted-by":"publisher","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, 1863\u20131893 (2007)","journal-title":"J. Comput. Phys."},{"key":"9962_CR16","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1016\/j.jmaa.2018.08.045","volume":"468","author":"Q Hong","year":"2018","unstructured":"Hong, Q., Wang, Y., Wang, J.: Optimal error estimate of a linear fourier pseudo-spectral scheme for two dimensional Klein-Gordon-Schr\u00f6Dinger equations. J. Math. Anal. Appl. 468, 817\u2013838 (2018)","journal-title":"J. Math. Anal. Appl."},{"key":"9962_CR17","first-page":"799","volume":"203","author":"S Wang","year":"2008","unstructured":"Wang, S., Zhang, L.: A class of conservative orthogonal spline collocation schemes for solving coupled Klein-Gordon-Schr\u00f6Dinger equations. Appl. Math. Comput. 203, 799\u2013812 (2008)","journal-title":"Appl. Math. Comput."},{"key":"9962_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-85729-148-6","volume-title":"Numerical methods for ordinary differential equations: initial value problems","author":"DF Griffiths","year":"2010","unstructured":"Griffiths, D.F., Higham, D.J.: Numerical methods for ordinary differential equations: initial value problems. Springer, Berlin (2010)"},{"issue":"1","key":"9962_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1093\/imanum\/7.1.1","volume":"7","author":"GJ Cooper","year":"1987","unstructured":"Cooper, G.J.: Stability of Runge-Kutta methods for trajectory problems. IMA J. Numer. Anal. 7(1), 1\u201313 (1987)","journal-title":"IMA J. Numer. Anal."},{"key":"9962_CR20","volume-title":"Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, vol. 31","author":"E Hairer","year":"2006","unstructured":"Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, vol. 31. Springer, Berlin (2006)"},{"key":"9962_CR21","doi-asserted-by":"publisher","first-page":"1461","DOI":"10.1090\/S0025-5718-97-00902-2","volume":"66","author":"MP Calvo","year":"1997","unstructured":"Calvo, M.P., Iserles, A., Zanna, A.: Numerical solution of isospectral flows. Math. Comput. 66, 1461\u20131486 (1997)","journal-title":"Math. Comput."},{"key":"9962_CR22","doi-asserted-by":"publisher","first-page":"691","DOI":"10.1016\/j.cma.2016.10.041","volume":"315","author":"X Yang","year":"2017","unstructured":"Yang, X., Ju, L.: Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model. Comput. Methods Appl. Mech. Engrg. 315, 691\u2013712 (2017)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"9962_CR23","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1016\/j.jcp.2016.12.025","volume":"333","author":"X Yang","year":"2017","unstructured":"Yang, X., Zhao, J., Wang, Q.: Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method. J. Comput. Phys. 333, 104\u2013127 (2017)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"9962_CR24","doi-asserted-by":"publisher","first-page":"B889","DOI":"10.1137\/17M1125005","volume":"40","author":"X Yang","year":"2018","unstructured":"Yang, X., Yu, H.: Efficient second order unconditionally stable schemes for a phase field moving contact line model using an invariant energy quadratization approach. SIAM J. Sci. Comput. 40(3), B889\u2013B914 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"9962_CR25","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1016\/j.jcp.2017.10.021","volume":"353","author":"J Shen","year":"2018","unstructured":"Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407\u2013416 (2018)","journal-title":"J. Comput. Phys."},{"issue":"5","key":"9962_CR26","doi-asserted-by":"publisher","first-page":"2895","DOI":"10.1137\/17M1159968","volume":"56","author":"J Shen","year":"2018","unstructured":"Shen, J., Xu, J.: Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J. Numer. Anal. 56(5), 2895\u20132912 (2018)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"9962_CR27","doi-asserted-by":"publisher","first-page":"474","DOI":"10.1137\/17M1150153","volume":"61","author":"J Shen","year":"2019","unstructured":"Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474\u2013506 (2019)","journal-title":"SIAM Rev."},{"key":"9962_CR28","doi-asserted-by":"publisher","first-page":"224","DOI":"10.1016\/j.aml.2019.02.002","volume":"94","author":"Y Gong","year":"2019","unstructured":"Gong, Y., Zhao, J.: Energy-stable Runge-Kutta schemes for gradient flow models using the energy quadratization approach. Appl. Math. Lett. 94, 224\u2013231 (2019)","journal-title":"Appl. Math. Lett."},{"key":"9962_CR29","doi-asserted-by":"publisher","first-page":"B135","DOI":"10.1137\/18M1213579","volume":"42","author":"Y Gong","year":"2020","unstructured":"Gong, Y., Zhao, J., Wang, Q.: Arbitrary high-order unconditionally energy stable schemes for thermodynamically consistent gradient flow models. SIAM J. Sci. Comput. 42, B135\u2013B156 (2020)","journal-title":"SIAM J. Sci. Comput."},{"key":"9962_CR30","doi-asserted-by":"publisher","first-page":"109610","DOI":"10.1016\/j.jcp.2020.109610","volume":"419","author":"Y Gong","year":"2020","unstructured":"Gong, Y., Zhao, J., Wang, Q.: Arbitrarily high-order linear energy stable schemes for gradient flow models. J. Comput. Phys. 419, 109610 (2020)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"9962_CR31","doi-asserted-by":"publisher","first-page":"1629","DOI":"10.1007\/s10915-019-01001-5","volume":"80","author":"C Jiang","year":"2019","unstructured":"Jiang, C., Cai, W., Wang, Y.: A linearly implicit and local energy-preserving scheme for the Sine-Gordon equation based on the invariant energy quadratization approach. J. Sci. Comput. 80(3), 1629\u20131655 (2019)","journal-title":"J. Sci. Comput."},{"key":"9962_CR32","first-page":"161","volume":"358","author":"H Li","year":"2019","unstructured":"Li, H., Hong, Q.: An efficient energy-preserving algorithm for the Lorentz force system. Appl. Math. Comput. 358, 161\u2013168 (2019)","journal-title":"Appl. Math. Comput."},{"key":"9962_CR33","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/j.apnum.2019.12.016","volume":"151","author":"C Jiang","year":"2020","unstructured":"Jiang, C., Wang, Y., Gong, Y.: Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. Appl. Numer. Math. 151, 85\u201397 (2020)","journal-title":"Appl. Numer. Math."},{"key":"9962_CR34","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1007\/s10915-021-01533-9","volume":"88","author":"X Li","year":"2021","unstructured":"Li, X., Gong, Y., Zhang, L.: Linear high-order energy-preserving schemes for the nonlinear Schr\u00f6Dinger equation with wave operator using the scalar auxiliary variable approach. J. Sci. Comput. 88, 20 (2021)","journal-title":"J. Sci. Comput."},{"key":"9962_CR35","first-page":"402","volume":"51","author":"I Fukuda","year":"1975","unstructured":"Fukuda, I., Tsutsumi, M.: On the Yukawa-coupled Klein-Gordon-Schr\u00f6Dinger equations in three space dimensions. Proc. Jpn. Acad. 51, 402\u2013405 (1975)","journal-title":"Proc. Jpn. Acad."},{"key":"9962_CR36","doi-asserted-by":"publisher","first-page":"2445","DOI":"10.1063\/1.532875","volume":"40","author":"B Wang","year":"1999","unstructured":"Wang, B., Lange, H.: Attractors for the Klein-Gordon-Schr\u00f6Dinger equation. J. Math. Phys. 40, 2445\u20132457 (1999)","journal-title":"J. Math. Phys."},{"key":"9962_CR37","doi-asserted-by":"publisher","first-page":"354","DOI":"10.1016\/j.jcp.2016.10.022","volume":"328","author":"Y Gong","year":"2017","unstructured":"Gong, Y., Wang, Q., Wang, Y., Cai, J.: A conservative Fourier pseudo-spectral method for the nonlinear Schr\u00f6Dinger equation. J. Comput. Phys. 328, 354\u2013370 (2017)","journal-title":"J. Comput. Phys."},{"key":"9962_CR38","doi-asserted-by":"crossref","unstructured":"Zhang, Z., Gong, Y., Zhao, J.: A remark on the invariant energy quadratization (IEQ) method for preserving the original energy dissipation laws, arXiv:2111.12920v1 (2021)","DOI":"10.3934\/era.2022037"},{"key":"9962_CR39","doi-asserted-by":"publisher","first-page":"107331","DOI":"10.1016\/j.aml.2021.107331","volume":"120","author":"J Zhao","year":"2021","unstructured":"Zhao, J.: A revisit of the energy quadratization method with a relaxation technique. Appl. Math. Lett. 120, 107331 (2021)","journal-title":"Appl. Math. Lett."},{"key":"9962_CR40","doi-asserted-by":"publisher","first-page":"110954","DOI":"10.1016\/j.jcp.2022.110954","volume":"456","author":"M Jiang","year":"2022","unstructured":"Jiang, M., Zhang, Z., Zhao, J.: Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation. J. Comput. Phys. 456, 110954 (2022)","journal-title":"J. Comput. Phys."},{"key":"9962_CR41","doi-asserted-by":"publisher","first-page":"113070","DOI":"10.1016\/j.cma.2020.113070","volume":"367","author":"Q Cheng","year":"2020","unstructured":"Cheng, Q., Liu, C., Shen, J.: A new Lagrange multiplier approach for gradient flows. Comput. Methods Appl. Mech. Engrg. 367, 113070 (2020)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"9962_CR42","doi-asserted-by":"publisher","first-page":"113746","DOI":"10.1016\/j.cma.2021.113746","volume":"381","author":"Y Gong","year":"2021","unstructured":"Gong, Y., Hong, Q., Wang, Q.: Supplementary variable method for thermodynamically consistent partial differential equations. Comput. Methods Appl. Mech. Engrg. 381, 113746 (2021)","journal-title":"Comput. Methods Appl. Mech. Engrg."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-022-09962-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-022-09962-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-022-09962-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,23]],"date-time":"2023-11-23T13:09:08Z","timestamp":1700744948000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-022-09962-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,21]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,8]]}},"alternative-id":["9962"],"URL":"https:\/\/doi.org\/10.1007\/s10444-022-09962-2","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":"31 August 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 May 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-->Competing interests"}}],"article-number":"41"}}