{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T20:47:21Z","timestamp":1760647641346,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,28]],"date-time":"2022-06-28T00:00:00Z","timestamp":1656374400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004763","name":"Natural Science Foundation of Inner Mongolia","doi-asserted-by":"publisher","award":["2020MS01003","2021MS01018","NMGIRT2207","202110126023"],"award-info":[{"award-number":["2020MS01003","2021MS01018","NMGIRT2207","202110126023"]}],"id":[{"id":"10.13039\/501100004763","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region","award":["2020MS01003","2021MS01018","NMGIRT2207","202110126023"],"award-info":[{"award-number":["2020MS01003","2021MS01018","NMGIRT2207","202110126023"]}]},{"name":"National Innovation Project","award":["2020MS01003","2021MS01018","NMGIRT2207","202110126023"],"award-info":[{"award-number":["2020MS01003","2021MS01018","NMGIRT2207","202110126023"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, the coupled Schr\u00f6dinger\u2013Boussinesq equations are solved numerically using the finite element method combined with the time two-mesh (TT-M) fast algorithm. The spatial direction is discretized by the standard Galerkin finite element method, the temporal direction is approximated by the TT-M Crank\u2013Nicolson scheme, and then the numerical scheme of TT-M finite element (FE) system is formulated. The method includes three main steps: for the first step, the nonlinear system is solved on the coarse time mesh; for the second step, by an interpolation formula, the numerical solutions at the fine time mesh point are computed based on the numerical solutions on the coarse mesh system; for the last step, the linearized temporal fine mesh system is constructed based on Taylor\u2019s formula for two variables, and then the TT-M FE solutions can be obtained. Furthermore, theory analyses on the TT-M system including the stability and error estimations are conducted. Finally, a large number of numerical examples are provided to verify the accuracy of the algorithm, the correctness of theoretical results, and the computational efficiency with a comparison to the numerical results calculated by using the standard FE method.<\/jats:p>","DOI":"10.3390\/axioms11070314","type":"journal-article","created":{"date-parts":[[2022,6,28]],"date-time":"2022-06-28T23:59:22Z","timestamp":1656460762000},"page":"314","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["TT-M Finite Element Algorithm for the Coupled Schr\u00f6dinger\u2013Boussinesq Equations"],"prefix":"10.3390","volume":"11","author":[{"given":"Jiale","family":"Tian","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"}]},{"given":"Ziyu","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8218-0196","authenticated-orcid":false,"given":"Yang","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9832-0496","authenticated-orcid":false,"given":"Hong","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1143\/PTP.62.370","article-title":"Soliton solutions in a diatomic lattice system","volume":"62","author":"Yajima","year":"1979","journal-title":"Prog. Theor. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF02846945","article-title":"Coupled scalar field equations for nonlinear wave modulations in dispersive media","volume":"46","author":"Rao","year":"1996","journal-title":"Pramana J. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.apnum.2017.04.007","article-title":"Numerical analysis of cubic orthogonal spline collocation methods for the coupled Schr\u00f6dinger-Boussinesq equations","volume":"119","author":"Liao","year":"2017","journal-title":"Appl. Numer. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/S1007-5704(01)90030-9","article-title":"The behavior of attractors for damped Schr\u00f6dinger-Boussinesq equation","volume":"6","author":"Guo","year":"2001","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_5","first-page":"517","article-title":"Application of the extended simplest equation method to the coupled Schr\u00f6dinger-Boussinesq equation","volume":"224","author":"Bilige","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/s10114-010-8034-6","article-title":"The global solution of the system of equations for complex Schr\u00f6dinger field coupled with Boussinesq type self-consistent field","volume":"26","author":"Guo","year":"1983","journal-title":"Acta Math. Sin."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1006\/jmaa.1996.5148","article-title":"Finite dimensional global attractor for dissipative Schr\u00f6dinger-Boussinesq equations","volume":"205","author":"Li","year":"1997","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0167-2789(95)00277-4","article-title":"Finite dimensional behavior of global attractors for weakly damped nonlinear Schr\u00f6dinger-Boussinesq equations","volume":"93","author":"Guo","year":"1996","journal-title":"Physica D"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1016\/j.jmaa.2010.03.007","article-title":"On the periodic Schr\u00f6dinger-Boussinesq system","volume":"368","author":"Farah","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1006\/jmaa.2000.7455","article-title":"Existence of the periodic solution for the weakly damped Schr\u00f6dinger-Boussinesq equation","volume":"262","author":"Guo","year":"2001","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3501","DOI":"10.1016\/j.na.2009.02.029","article-title":"A series of exact solutions for coupled Higgs field equation and coupled Schr\u00f6dinger-Boussinesq equation","volume":"71","author":"Hon","year":"2009","journal-title":"Nonlinear Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4813","DOI":"10.1088\/0305-4470\/22\/22\/012","article-title":"Exact solutions of coupled scalar field equations","volume":"22","author":"Rao","year":"1989","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/S0252-9602(17)30488-5","article-title":"Exact explicit solutions of the nonlinear Schr\u00f6dinger equation coupled to the Boussinesq equation","volume":"23","author":"Xia","year":"2003","journal-title":"Acta Math. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.cnsns.2017.06.033","article-title":"Time-splitting combined with exponential wave integrator Fourier pseudospectral method for Schr\u00f6dinger-Boussinesq system","volume":"55","author":"Liao","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simulat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4899","DOI":"10.1016\/j.cam.2011.04.001","article-title":"Numerical analysis for a conservative difference scheme to solve the Schr\u00f6dinger-Boussinesq equation","volume":"235","author":"Zhang","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1667","DOI":"10.1002\/num.22067","article-title":"Conservative compact finite difference scheme for the coupled Schr\u00f6dinger-Boussinesq equation","volume":"32","author":"Liao","year":"2016","journal-title":"Numer. Meth. Part. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.apnum.2021.07.013","article-title":"Analysis of the linearly energy- and mass-preserving finite difference methods for the coupled Schr\u00f6dinger-Boussinesq equations","volume":"170","author":"Deng","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"ref_18","first-page":"133","article-title":"The finite element analysis for the equation system coupling the complex Schr\u00f6dinger and real Boussinesq fields","volume":"9","author":"Zheng","year":"1987","journal-title":"Math. Numer. Sin."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1714","DOI":"10.1080\/00207160.2010.522234","article-title":"The quadratic B-spline finite element method for the coupled Schr\u00f6dinger-Boussinesq equations","volume":"88","author":"Bai","year":"2011","journal-title":"Int. J. Comput. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"070201","DOI":"10.1088\/1674-1056\/22\/7\/070201","article-title":"Multi-symplectic scheme for the coupled Schr\u00f6dinger-Boussinesq equations","volume":"7","author":"Huang","year":"2013","journal-title":"Chin. Phys. B"},{"key":"ref_21","first-page":"1","article-title":"Efficient energy-preserving wavelet collocation schemes for the coupled nonlinear Schr\u00f6dinger-Boussinesq system","volume":"357","author":"Cai","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_22","first-page":"344","article-title":"The convergence of Galerkin-Fourier method for equation of Schr\u00f6dinger-Boussinesq field","volume":"2","author":"Guo","year":"1984","journal-title":"J. Comput. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1132","DOI":"10.1016\/j.ijheatmasstransfer.2017.12.118","article-title":"Time two-mesh algorithm combined with finite element method for time fractional water wave model","volume":"120","author":"Liu","year":"2018","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/j.jcp.2018.12.004","article-title":"Fast algorithm based on TT-M FE system for space fractional Allen-Cahn equations with smooth and non-smooth solutions","volume":"379","author":"Yin","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1793","DOI":"10.1016\/j.camwa.2020.08.011","article-title":"TT-M finite element algorithm for a two-dimensional space fractional Gray-Scott model","volume":"80","author":"Liu","year":"2020","journal-title":"Comput. Math. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1007\/s11075-020-01048-8","article-title":"Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model","volume":"88","author":"Wen","year":"2021","journal-title":"Numer. Algorithms"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2417","DOI":"10.1007\/s40840-019-00810-z","article-title":"A fast time two-mesh algorithm for Allen-Cahn equation","volume":"43","author":"Wang","year":"2020","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s11075-019-00801-y","article-title":"A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile\/immobile transport model","volume":"85","author":"Qiu","year":"2020","journal-title":"Numer. Algorithms"},{"key":"ref_29","unstructured":"Niu, Y.X., Liu, Y., Li, H., and Liu, F.W. (2021). Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media, submitted."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Tutueva, A., Karimov, T., and Butusov, D. (2020). Semi-implicit and semi-explicit Adams-Bashforth-Moulton methods. Mathematics, 8.","DOI":"10.3390\/math8050780"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Tutueva, A., and Butusov, D. (2021). Avoiding dynamical degradation in computer simulation of chaotic systems using semi-explicit integration: R\u00f6ssler oscillator case. Fract. Fract., 5.","DOI":"10.3390\/fractalfract5040214"},{"key":"ref_32","unstructured":"Thomee, V. (1984). Galerkin Finite Element Method for Parabolic Problems, Springer."},{"key":"ref_33","unstructured":"Wen, C., Wang, J.F., Liu, Y., Li, H., and Fang, Z.C. (2022). Unconditionally optimal time two-mesh mixed finite element algorithm for a nonlinear distributed-order fourth-order equation, submitted."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1007\/s11075-016-0160-5","article-title":"Galerkin finite element method for nonlinear fractional Schr\u00f6dinger equations","volume":"74","author":"Li","year":"2017","journal-title":"Numer. Algorithms"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"A3067","DOI":"10.1137\/16M1105700","article-title":"Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional Schr\u00f6dinger equations","volume":"39","author":"Li","year":"2017","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"109869","DOI":"10.1016\/j.jcp.2020.109869","article-title":"A structure preserving difference scheme with fast algorithms for high dimensional nonlinear space-fractional Schr\u00f6dinger equations","volume":"425","author":"Yin","year":"2021","journal-title":"J. Comput. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1439","DOI":"10.1080\/00207160.2014.945440","article-title":"Finite difference method for time-space-fractional Schr\u00f6dinger equation","volume":"92","author":"Liu","year":"2015","journal-title":"Int. J. Comput. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.cam.2019.01.045","article-title":"Linearized Crank-Nicolson scheme for the nonlinear time-space fractional Schr\u00f6dinger equations","volume":"355","author":"Ran","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_39","first-page":"124689","article-title":"The global analysis on the spectral collocation method for time fractional Schr\u00f6dinger equation","volume":"365","author":"Zheng","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.apnum.2018.10.012","article-title":"Split-step spectral Galerkin method for the two-dimensional nonlinear space-fractional Schr\u00f6dinger equation","volume":"136","author":"Wang","year":"2019","journal-title":"Appl. Numer. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1080\/00207160.2018.1434515","article-title":"Crank-Nicolson Fourier spectral methods for the space fractional nonlinear Schr\u00f6dinger equation and its parameter estimation","volume":"96","author":"Zhang","year":"2019","journal-title":"Int. J. Comput. Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"644","DOI":"10.1016\/j.jcp.2014.04.047","article-title":"A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schr\u00f6dinger equations","volume":"272","author":"Wang","year":"2014","journal-title":"J. Comput. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.matcom.2021.02.012","article-title":"Galerkin approximation with quintic B-spline as basis and weight functions for solving second order coupled nonlinear Schr\u00f6dinger equations","volume":"187","author":"Iqbal","year":"2021","journal-title":"Math. Comput. Simulat."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.camwa.2010.11.007","article-title":"High-order compact splitting multisymplectic method for the coupled nonlinear Schr\u00f6dinger equations","volume":"61","author":"Ma","year":"2011","journal-title":"Comput. Math. Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/7\/314\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:39:50Z","timestamp":1760139590000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/7\/314"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,28]]},"references-count":44,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["axioms11070314"],"URL":"https:\/\/doi.org\/10.3390\/axioms11070314","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,6,28]]}}}