{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,1,7]],"date-time":"2023-01-07T06:07:57Z","timestamp":1673071677655},"reference-count":30,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["NHM"],"published-print":{"date-parts":[[2023]]},"abstract":"<jats:p xml:lang=\"fr\">&lt;abstract&gt;&lt;p&gt;In this paper, we design a novel class of arbitrarily high-order, linearly implicit and energy-preserving numerical schemes for solving the nonlinear dispersive equations. Based on the idea of the energy quadratization technique, the original system is firstly rewritten as an equivalent system with a quadratization energy. The prediction-correction strategy, together with the Partitioned Runge-Kutta method, is then employed to discretize the reformulated system in time. The resulting semi-discrete system is high-order, linearly implicit and can preserve the quadratic energy of the reformulated system exactly. Finally, we take the Camassa-Holm equation as a benchmark to show the efficiency and accuracy of the proposed schemes.&lt;\/p&gt;&lt;\/abstract&gt;<\/jats:p>","DOI":"10.3934\/nhm.2023016","type":"journal-article","created":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T10:49:19Z","timestamp":1673002159000},"page":"399-411","source":"Crossref","is-referenced-by-count":0,"title":["A high-order linearly implicit energy-preserving Partitioned Runge-Kutta scheme for a class of nonlinear dispersive equations"],"prefix":"10.3934","volume":"18","author":[{"given":"Jin","family":"Cui","sequence":"first","affiliation":[{"name":"Department of Basic Sciences, Nanjing Vocational College of Information Technology, Nanjing 210023, China"}]},{"given":"Yayun","family":"Fu","sequence":"additional","affiliation":[{"name":"School of Science, Xuchang University, Henan Joint International Research Laboratory of High Performance Computation for Complex Systems, Xuchang 461000, China"}]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2023016-1","doi-asserted-by":"publisher","unstructured":"L. Brugnano, F. Iavernaro, J. Montijano, L. R$\\rm \\acute{a}$ndez, Spectrally accurate space-time solution of Hamiltonian PDEs, <i>Numer. Algorithms.<\/i>, <b>81<\/b> (2019), 1183\u20131202. https:\/\/doi.org\/10.1007\/s11075-018-0586-z","DOI":"10.1007\/s11075-018-0586-z"},{"key":"key-10.3934\/nhm.2023016-2","doi-asserted-by":"publisher","unstructured":"R. Camassa, D. D. Holm, A integrable shallow water equation with peaked solutions, <i>Phys. Rev. Lett.<\/i>, <b>71<\/b> (1993), 1661\u20131664. https:\/\/doi.org\/10.1103\/PhysRevLett.71.1661","DOI":"10.1103\/PhysRevLett.71.1661"},{"key":"key-10.3934\/nhm.2023016-3","doi-asserted-by":"publisher","unstructured":"R. Camassa, D. D. Holm, J. M. Hyman, A new integrable shallow water equation, <i>Adv. Appl. Mech.<\/i>, <b>31<\/b> (1994), 1\u201333. https:\/\/doi.org\/10.1093\/rheumatology\/33.1.31","DOI":"10.1093\/rheumatology\/33.1.31"},{"key":"key-10.3934\/nhm.2023016-4","doi-asserted-by":"publisher","unstructured":"D. Cohen, B. Owren, X. Raynaud, Multi-symplectic integration of the Camassa-Holm equation, <i>J. Comput. Phys.<\/i>, <b>227<\/b> (2008), 5492\u20135512. https:\/\/doi.org\/10.1016\/j.jcp.2008.01.051","DOI":"10.1016\/j.jcp.2008.01.051"},{"key":"key-10.3934\/nhm.2023016-5","doi-asserted-by":"publisher","unstructured":"D. Cohen, X. Raynaud, Geometric finite difference schemes for the generalized hyperelastic-rod wave equation, <i>J. Comput. Appl. Math.<\/i>, <b>235<\/b> (2011), 1925\u20131940. https:\/\/doi.org\/10.1016\/j.cam.2010.09.015","DOI":"10.1016\/j.cam.2010.09.015"},{"key":"key-10.3934\/nhm.2023016-6","doi-asserted-by":"publisher","unstructured":"S. Eidnes, L. Li, Linearly implicit local and global energy-preserving methods for PDEs with a cubic Hamiltonian, <i>SIAM J. Sci. Comput.<\/i>, <b>42<\/b> (2020), A2865\u2013A2888. https:\/\/doi.org\/10.1137\/19M1272688","DOI":"10.1137\/19M1272688"},{"key":"key-10.3934\/nhm.2023016-7","doi-asserted-by":"publisher","unstructured":"S. Eidnes, L. Li, S. Sato, Linearly implicit structure-preserving schemes for Hamiltonian systems, <i>J. Comput. Appl. Math.<\/i>, <b>387<\/b> (2021), 112489. https:\/\/doi.org\/10.1016\/j.cam.2019.112489","DOI":"10.1016\/j.cam.2019.112489"},{"key":"key-10.3934\/nhm.2023016-8","doi-asserted-by":"crossref","unstructured":"D. Furihata, T. Matsuo, <i>Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations<\/i>, London: Chapman &amp; Hall\/CRC, 2011.","DOI":"10.1201\/b10387"},{"key":"key-10.3934\/nhm.2023016-9","doi-asserted-by":"publisher","unstructured":"Y. Gong, Y. Wang, An energy-preserving wavelet collocation method for general multi-symplectic formulations of Hamiltonian PDEs, <i>Commun. Comput. Phys.<\/i>, <b>20<\/b> (2016), 1313\u20131339. https:\/\/doi.org\/10.4208\/cicp.231014.110416a","DOI":"10.4208\/cicp.231014.110416a"},{"key":"key-10.3934\/nhm.2023016-10","doi-asserted-by":"publisher","unstructured":"Y. Gong, J. Zhao, Energy-stable Runge-Kutta schemes for gradient flow models using the energy quadratization approach, <i>Appl. Math. Lett.<\/i>, <b>94<\/b> (2019), 224\u2013231. https:\/\/doi.org\/10.1016\/j.aml.2019.02.002","DOI":"10.1016\/j.aml.2019.02.002"},{"key":"key-10.3934\/nhm.2023016-11","doi-asserted-by":"publisher","unstructured":"Y. Gong, J. Zhao, Q. Wang, Arbitrarily high-order linear energy stable schemes for gradient flow models, <i>J. Comput. Phys.<\/i>, <b>419<\/b> (2020), 109610. https:\/\/doi.org\/10.1016\/j.jcp.2020.109610","DOI":"10.1016\/j.jcp.2020.109610"},{"key":"key-10.3934\/nhm.2023016-12","unstructured":"E. Hairer, C. Lubich, G. Wanner, <i>Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations<\/i>, Berlin: Springer-Verlag, 2006."},{"key":"key-10.3934\/nhm.2023016-13","doi-asserted-by":"publisher","unstructured":"Q. Hong, Y. Gong, Z. Lv, Linear and Hamiltonian-conserving Fourier pseudo-spectral schemes for the Camassa-Holm equation, <i>Appl. Math. Comput.<\/i>, <b>346<\/b> (2019), 86\u201395. https:\/\/doi.org\/10.1016\/j.amc.2018.10.043","DOI":"10.1016\/j.amc.2018.10.043"},{"key":"key-10.3934\/nhm.2023016-14","doi-asserted-by":"crossref","unstructured":"C. Jiang, Y. Gong, W. Cai, Y. Wang, A linearly implicit structure-preserving scheme for the Camassa-Holm equation based on multiple scalar auxiliary variables approach, <i>J. Sci. Comput.<\/i>, <b>83<\/b> (2020), 1\u201320.","DOI":"10.1007\/s10915-020-01201-4"},{"key":"key-10.3934\/nhm.2023016-15","doi-asserted-by":"publisher","unstructured":"C. Jiang, Y. Wang, Y. Gong, Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation, <i>Appl. Numer. Math.<\/i>, <b>151<\/b> (2020), 85\u201397. https:\/\/doi.org\/10.1016\/j.apnum.2019.12.016","DOI":"10.1016\/j.apnum.2019.12.016"},{"key":"key-10.3934\/nhm.2023016-16","doi-asserted-by":"crossref","unstructured":"H. Liu, T. Pendleton, On invariant-preserving finite difference schemes for the Camassa-Holm equation and the two-component Camassa-Holm system, <i>Commun. Comput. Phys.<\/i>, <b>19<\/b> (2016), 1015\u20131041.","DOI":"10.4208\/cicp.130115.110915a"},{"key":"key-10.3934\/nhm.2023016-17","doi-asserted-by":"publisher","unstructured":"T. Matsuo, A Hamiltonian-conserving Galerkin scheme for the Camassa-Holm equation, <i>J. Comput. Appl. Math.<\/i>, <b>234<\/b> (2010), 1258\u20131266. https:\/\/doi.org\/10.1016\/j.cam.2009.09.020","DOI":"10.1016\/j.cam.2009.09.020"},{"key":"key-10.3934\/nhm.2023016-18","doi-asserted-by":"publisher","unstructured":"T. Matsuo, H. Yamaguchi, An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations, <i>J. Comput. Phys.<\/i>, <b>228<\/b> (2009), 4346\u20134358. https:\/\/doi.org\/10.1016\/j.jcp.2009.03.003","DOI":"10.1016\/j.jcp.2009.03.003"},{"key":"key-10.3934\/nhm.2023016-19","doi-asserted-by":"crossref","unstructured":"Y. Miyatake, T. Matsuo, A general framework for finding energy dissipative\/conservative $H^1$-Galerkin schemes and their underlying $H^1$-weak forms for nonlinear evolution equations, <i>BIT<\/i>, <b>54<\/b> (2014), 1119\u20131154.","DOI":"10.1007\/s10543-014-0483-3"},{"key":"key-10.3934\/nhm.2023016-20","doi-asserted-by":"crossref","unstructured":"B. N. Ryland, R. I. McLachlan, On multisymplecticity of Partitioned Runge-Kutta methods, <i>SIAM J. Sci. Comput.<\/i>, <b>30<\/b> (2008), 1318\u20131340.","DOI":"10.1137\/070688468"},{"key":"key-10.3934\/nhm.2023016-21","doi-asserted-by":"crossref","unstructured":"J. Shen, J. Xu, Stabilized predictor-corrector schemes for gradient flows with strong anisotropic free energy, <i>Commun. Comput. Phys.<\/i>, <b>24<\/b> (2018), 635\u2013654.","DOI":"10.4208\/cicp.OA-2017-0209"},{"key":"key-10.3934\/nhm.2023016-22","doi-asserted-by":"publisher","unstructured":"J. Shen, J. Xu, J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, <i>SIAM Rev.<\/i>, <b>61<\/b> (2019), 474\u2013506. https:\/\/doi.org\/10.1137\/070688468","DOI":"10.1137\/070688468"},{"key":"key-10.3934\/nhm.2023016-23","unstructured":"G. Sun, Symplectic partitioned Runge-Kutta methods, <i>J. Comput. Math.<\/i>, <b>4<\/b> (1993), 365\u2013372."},{"key":"key-10.3934\/nhm.2023016-24","doi-asserted-by":"crossref","unstructured":"Z. Sun, Y. Xing, On structure-preserving discontinuous Galerkin methods for Hamiltonian partial differential equations: energy conservation and multi-symplecticity, <i>J. Comput. Phys.<\/i>, <b>419<\/b> (2020), 109662.","DOI":"10.1016\/j.jcp.2020.109662"},{"key":"key-10.3934\/nhm.2023016-25","unstructured":"G. B. Whitham, <i>Linear and Nonlinear Waves<\/i>. New York: John Wiley &amp; Sons, 1974."},{"key":"key-10.3934\/nhm.2023016-26","doi-asserted-by":"publisher","unstructured":"Y. Xu, C. W. Shu, A local discontinuous Galerkin method for the Camassa-Holm equation, <i>SIAM J. Numer. Anal.<\/i>, <b>46<\/b> (2008), 1998\u20132021. https:\/\/doi.org\/10.1137\/070679764","DOI":"10.1137\/070679764"},{"key":"key-10.3934\/nhm.2023016-27","doi-asserted-by":"publisher","unstructured":"X. Yang, J. Zhao, Q. Wang, Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method, <i>J. Comput. Phys.<\/i>, <b>333<\/b> (2017), 104\u2013127. https:\/\/doi.org\/10.1016\/j.jcp.2016.12.025","DOI":"10.1016\/j.jcp.2016.12.025"},{"key":"key-10.3934\/nhm.2023016-28","doi-asserted-by":"publisher","unstructured":"Z. Zhang, Y. Gong, J. Zhao, A remark on the invariant energy quadratization (IEQ) method for preserving the original energy dissipation laws, <i>Electron. Res. Arch.<\/i>, <b>30<\/b> (2022), 701\u2013714. https:\/\/doi.org\/10.3934\/era.2022037","DOI":"10.3934\/era.2022037"},{"key":"key-10.3934\/nhm.2023016-29","doi-asserted-by":"publisher","unstructured":"J. Zhao, A revisit of the energy quadratization method with a relaxation technique, <i>Appl. Math. Lett.<\/i>, <b>120<\/b> (2021), 107331. https:\/\/doi.org\/10.1016\/j.aml.2021.107331","DOI":"10.1016\/j.aml.2021.107331"},{"key":"key-10.3934\/nhm.2023016-30","doi-asserted-by":"crossref","unstructured":"H. Zhu, S. Song, Y. Tang, Multi-symplectic wavelet collocation method for the nonlinear Schr\u00f6dinger equation and the Camassa-Holm equation, <i>Comput. Phys. Commun.<\/i>, <b>182<\/b> (2011), 616\u2013627.","DOI":"10.1016\/j.cpc.2010.11.003"}],"container-title":["Networks and Heterogeneous Media"],"original-title":[],"link":[{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2023016?viewType=html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T10:49:30Z","timestamp":1673002170000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2023016"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023]]},"references-count":30,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023]]}},"URL":"https:\/\/doi.org\/10.3934\/nhm.2023016","relation":{},"ISSN":["1556-1801"],"issn-type":[{"value":"1556-1801","type":"print"}],"subject":[],"published":{"date-parts":[[2023]]}}}