{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:22:37Z","timestamp":1760145757401,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,23]],"date-time":"2024-08-23T00:00:00Z","timestamp":1724371200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007129","name":"Natural Science Foundations of Shandong","doi-asserted-by":"publisher","award":["ZR2023MA062","ZR202204010001","2021dzkj1638","2023XKZX024"],"award-info":[{"award-number":["ZR2023MA062","ZR202204010001","2021dzkj1638","2023XKZX024"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science and Technology Plan Project of Dezhou","award":["ZR2023MA062","ZR202204010001","2021dzkj1638","2023XKZX024"],"award-info":[{"award-number":["ZR2023MA062","ZR202204010001","2021dzkj1638","2023XKZX024"]}]},{"name":"Research Platform Project of Dezhou University","award":["ZR2023MA062","ZR202204010001","2021dzkj1638","2023XKZX024"],"award-info":[{"award-number":["ZR2023MA062","ZR202204010001","2021dzkj1638","2023XKZX024"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Good Boussinesq equations are considered in this work. First, we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then, three fully discrete schemes are developed based on a symplectic scheme in the time direction, which preserves the symplectic structure. Meanwhile, the convergence and conservation of the fully discrete schemes are analyzed. Finally, we present numerical experiments to confirm our theoretical analysis. Both our analysis and numerical tests indicate that the fully discrete schemes are efficient in solving the spatial derivative mixed equation.<\/jats:p>","DOI":"10.3390\/axioms13090574","type":"journal-article","created":{"date-parts":[[2024,8,23]],"date-time":"2024-08-23T12:58:07Z","timestamp":1724417887000},"page":"574","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Combined Compact Symplectic Schemes for the Solution of Good Boussinesq Equations"],"prefix":"10.3390","volume":"13","author":[{"given":"Zhenyu","family":"Lang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4495-4098","authenticated-orcid":false,"given":"Xiuling","family":"Yin","sequence":"additional","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanqin","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiguo","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shuxia","family":"Kong","sequence":"additional","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"946","DOI":"10.1137\/0905065","article-title":"Numerical solutions of the good Boussinesq equation","volume":"5","author":"Manotanjan","year":"1984","journal-title":"Siam. J. Sci. Stat. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1964","DOI":"10.1063\/1.527850","article-title":"Soliton and antisoliton interactions in the \u201cgood\u201d Boussinesq equation","volume":"1988 29","author":"Manotanjan","year":"1988","journal-title":"J. Math. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/BF01385620","article-title":"Nonlinear stability and convergence of finite-difference methods for the \u201cgood\u201d Boussinesq equation","volume":"58","author":"Ortega","year":"1990","journal-title":"Numer. Math."},{"key":"ref_4","first-page":"1","article-title":"Complex structure-preserving method for Schrodinger equations in quaternionic quantum mechanics","volume":"150","author":"Guo","year":"2023","journal-title":"Numer. Algorithms"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Iqbal, A., Abd Hamid, N.N., and Ismail, A.I.M. (2019). Soliton Solution of Schr\u00f6dinger Equation Using Cubic B-Spline Galerkin Method. Fluids, 4.","DOI":"10.3390\/fluids4020108"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.matcom.2020.02.017","article-title":"Cubic B-spline Galerkin method for numerical solution of the coupled nonlinear Schr\u00f6dinger equation","volume":"174","author":"Iqbal","year":"2020","journal-title":"Math. Comput. Sim."},{"key":"ref_7","first-page":"5349","article-title":"Algebraic algorithms for a class of Schrodinger equations in split quaternionic mechanics","volume":"47","author":"Jiang","year":"2024","journal-title":"Math. Math. Methods. Appl. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1124","DOI":"10.4208\/jcm.2302-m2022-0033","article-title":"Two-grid finite element method for time-fractional nonlinear schrodinger equation","volume":"42","author":"Hu","year":"2024","journal-title":"J. Comput. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"108975","DOI":"10.1016\/j.aml.2023.108975","article-title":"A fast algorithm for the Schr\u00f6dinger equation in quaternionic quantum mechanics","volume":"150","author":"Jiang","year":"2024","journal-title":"Appl. Math. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2217","DOI":"10.1016\/j.camwa.2017.12.006","article-title":"Algebraic techniques for Schrodinger equations in split quaternionic mechanics","volume":"75","author":"Jiang","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1007\/s10915-023-02219-0","article-title":"Structure-Preserving Combined High-Order Compact Schemes for Multiple Order Spatial Derivatives Differential Equations","volume":"96","author":"Wang","year":"2023","journal-title":"J. Sci. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4151","DOI":"10.1016\/j.apm.2010.04.012","article-title":"A numerical technique for solution of the MRLW equation using quartic B-splines","volume":"34","author":"Haq","year":"2010","journal-title":"Appl. Math. Model"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1006\/jcph.1998.5899","article-title":"A three-point combined compact difference scheme","volume":"140","author":"Chu","year":"1998","journal-title":"J. Comput. Phys."},{"key":"ref_14","first-page":"125202","article-title":"Highly accurate compact difference scheme for fourth order parabolic equation with Dirichlet and Neumann boundary conditions","volume":"378","author":"Kaur","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"156","DOI":"10.11648\/j.sd.20160402.27","article-title":"Containing High Order Compact Scheme Source of Steady Convection-Diffusion Equation","volume":"4","author":"Wang","year":"2016","journal-title":"Sci. Discov."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"299","DOI":"10.4208\/nmtma.OA-2017-0090","article-title":"A linearized high-order combined compact difference scheme for multidimensional coupled Burgers equations","volume":"11","author":"Chen","year":"2018","journal-title":"Numer. Math. Theor. Methods. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.jcp.2015.07.051","article-title":"An optimized dispersion\u2013relation-preserving combined compact difference scheme to solve advection equations","volume":"300","author":"Yu","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.apnum.2017.04.006","article-title":"A second order operator splitting numerical scheme for the \u201cgood\u201d Boussinesq equation","volume":"119","author":"Zhang","year":"2017","journal-title":"Appl. Numer. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1743","DOI":"10.1002\/mma.4696","article-title":"Efficient structure-preserving schemes for good Boussinesq equation","volume":"41","author":"Chen","year":"2018","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"927","DOI":"10.1007\/s10483-008-0711-3","article-title":"Multi-symplectic method for generalized Boussinesq equation","volume":"29","author":"Hu","year":"2008","journal-title":"Appl. Math. Mech."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1423","DOI":"10.1002\/fld.1359","article-title":"A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part II: Boundary conditions and validation","volume":"53","author":"Cienfuegos","year":"2007","journal-title":"Int. J. Numer. Methods Fluids."},{"key":"ref_22","first-page":"109","article-title":"Pseudospectral method for the \u201cgood\u201d Boussinesq equation","volume":"57","author":"Frutos","year":"1991","journal-title":"Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4047","DOI":"10.1016\/j.jde.2013.02.006","article-title":"Improved local well-posedness for the periodic \u201cgood\u201d Boussinesq equation","volume":"254","author":"Oh","year":"2013","journal-title":"J. Differ. Equ."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1090\/S0002-9939-09-10142-9","article-title":"On the periodic \u201cgood\u201d Boussinesq equation","volume":"138","author":"Farah","year":"2010","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1016\/S0029-8018(98)00019-5","article-title":"Higher order Boussinesq equations","volume":"26","author":"Zou","year":"1999","journal-title":"Ocean Eng."},{"key":"ref_26","first-page":"1612","article-title":"Numerical solutions of Boussinesq equation using Galerkin finite element method","volume":"37","author":"Ucar","year":"2020","journal-title":"Wiley"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1016\/j.apnum.2018.09.005","article-title":"A novel kind of efficient symplectic scheme for Klein-Gordon-Schr\u00f6dinger equation","volume":"135","author":"Kong","year":"2019","journal-title":"Appl. Numer. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/0021-9991(92)90324-R","article-title":"Compact finite difference schemes with spectral-like solution","volume":"103","author":"Lele","year":"1992","journal-title":"J. Comput. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/S0375-9601(01)00294-8","article-title":"Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity","volume":"284","author":"Bridges","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1137\/17M1150153","article-title":"A new class of efficient and robust energy stable schemes for gradient flows","volume":"61","author":"Shen","year":"2019","journal-title":"SIAM Rev."},{"key":"ref_31","unstructured":"Jim\u00e9nez, S., and V\u00e1zquez., L. (1991). Some Remarks on Conservative and Symplectic Schemes, World Scientific."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/574\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:42:24Z","timestamp":1760110944000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/574"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,23]]},"references-count":31,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["axioms13090574"],"URL":"https:\/\/doi.org\/10.3390\/axioms13090574","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,8,23]]}}}