Skip to main content Accessibility help
×
Home
Hostname: page-component-5959bf8d4d-9mpts Total loading time: 0.285 Render date: 2022-12-10T10:26:58.685Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

LINEARLY IMPLICIT ENERGY-PRESERVING FOURIER PSEUDOSPECTRAL SCHEMES FOR THE COMPLEX MODIFIED KORTEWEG–DE VRIES EQUATION

Published online by Cambridge University Press:  12 January 2021

J. L. YAN*
Affiliation:
Department of Mathematics and Computer, Wuyi University, Wu Yi Shan354300, China.
L. H. ZHENG
Affiliation:
Department of Information and Computer Technology, No. 1 Middle School of Nanping, Nanping353000, China; e-mail: 413845939@qq.com.
L. ZHU
Affiliation:
Department of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang212003, China; e-mail: 38196700@qq.com.
F. Q. LU
Affiliation:
Changzhou Institute of Technology, Changzhou213032, China; e-mail: 724075305@qq.com.

Abstract

We propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes.

Type
Research Article
Copyright
© Australian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cai, J. X. and Miao, J., “New explicit multisymplectic scheme for the complex modified Korteweg–de Vries equation”, Chin. Phys. Lett. 29 (2012) 030201; doi:10.1088/0256-307X/29/3/030201.CrossRefGoogle Scholar
Cai, W. J., Jiang, C. L. and Wang, Y. S., “Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions”, J. Comput. Phys. 395 (2019) 166185; doi:10.1016/j.jcp.2019.05.048.CrossRefGoogle Scholar
Calvo, M. P., De Frutos, J. and Novo, J., “Linearly implicit Runge–Kutta methods for advection–reaction–diffusion equations”, Appl. Numer. Math. 37 (2001) 535549; doi:10.1016/S0168-9274(00)00061-1.CrossRefGoogle Scholar
Chen, J. B. and Qin, M. Z., “Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation”, Electron. Trans. Numer. Anal. 12 (2001) 193204, available at http://etna.mcs.kent.edu/volumes/2001-2010/vol12/.Google Scholar
Erbay, S. and Suhubi, E. S., “Nonlinear wave propagation in micropolar media II. Special cases, solitary waves and Painlevé analysis”, Internat. J. Engrg. Sci. 27 (1989) 915919; doi:10.1016/0020-7225(89)90032-3.CrossRefGoogle Scholar
Furihata, D. and Matsuo, T., Discrete variational derivative method: a structure-preserving numerical method for partial differential equations, (CRC Press, London, 2010).CrossRefGoogle Scholar
Gong, Y. Z. and Zhao, J., “Energy-stable Runge–Kutta schemes for gradient flow models using the energy quadratization approach”, Appl. Math. Lett. 94 (2019) 224231; doi:10.1016/j.aml.2019.02.002.CrossRefGoogle Scholar
Gong, Y. Z., Zhao, J. and Wang, Q., “Arbitrarily high-order unconditionally energy stable SAV schemes for gradient flow models”, Comput. Phys. Comm. 249 (2020) 107033; doi:10.1016/j.cpc.2019.107033.CrossRefGoogle Scholar
Gong, Y. Z., Zhao, J. and Wang, Q., “Arbitrarily high-order unconditionally energy stable schemes for thermodynamically consistent gradient flow models”, SIAM J. Sci. Comput. 42 (2020) B135B156; doi:10.1137/18M1213579.CrossRefGoogle Scholar
Gong, Y. Z., Zhao, J., Yang, X. F. and Wang, Q., “Fully discrete second-order linear schemes for hydrodynamic phase field models of binary viscous fluid flows with variable densities”, SIAM J. Sci. Comput. 40 (2018) B138B167; doi:10.1137/17M1111759.CrossRefGoogle Scholar
Gorbacheva, O. B. and Ostrovsky, L. A., “Nonlinear vector waves in a mechanical model of a molecular chain”, Physica D 8 (1983) 223228; doi:10.1016/0167-2789(83)90319-6.CrossRefGoogle Scholar
Hong, Q., Gong, Y. Z. and Lv, Z. Q., “Linear and Hamiltonian-conserving Fourier pseudo-spectral schemes for the Camassa–Holm equation”, Appl. Math. Comput. 346 (2019) 8695; doi:10.1016/j.amc.2018.10.043.Google Scholar
Ismail, M. S., “Numerical solution of complex modified Korteweg–de Vries equation by Petrov–Galerkin method”, Appl. Math. Comput. 202 (2008) 520531; doi:10.1016/j.amc.2008.02.033.Google Scholar
Ismail, M. S., “Numerical solution of complex modified Korteweg–de Vries equation by collocation method”, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 749759; doi:10.1016/j.cnsns.2007.12.005.CrossRefGoogle Scholar
Karney, C. F. F., Sen, A. and Chu, F. Y. F., “Nonlinear evolution of lower hybrid waves”, Phys. Fluids 22 (1979) 940952; doi:10.1063/1.862688.CrossRefGoogle Scholar
Korkmaz, A. and Dağ, İ., “Solitary wave simulations of complex modified Korteweg–de Vries equation using differential quadrature method”, Comput. Phys. Comm. 180 (2009) 15161523; doi:10.1016/j.cpc.2009.04.012.CrossRefGoogle Scholar
Korostil, I. A. and Clarke, S. R., “Fourth-order numerical methods for the coupled Korteweg–de Vries equations”, ANZIAM J. 56 (2014), 275285; doi:10.1017/S1446181115000012.CrossRefGoogle Scholar
Muslu, G. M. and Erabay, H. A., “A split-step Fourier method for the complex modified Korteweg–de Vries equation”, Comput. Math. Appl. 45(2003) 503514; doi:10.1016/S0898-1221(03)80033-0.CrossRefGoogle Scholar
Uddina, M., Haqa, S. and Islam, S. U., “Numerical solution of complex modified Korteweg–de Vries equation by mesh-free collocation method”, Comput. Math. Appl. 58 (2009) 566578; doi:10.1016/j.camwa.2009.03.104.CrossRefGoogle Scholar
Yan, J. L. and Zheng, L. H., “Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation”, Int. J. Appl. Math. Comput. Sci. 27 (2017) 515525; doi:10.1515/amcs-2017-0036.CrossRefGoogle Scholar
Yang, X. F., Zhao, J. and Wang, Q., “Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method”, J. Comput. Phys. 333 (2017) 104127; doi:10.1016/j.jcp.2016.12.025.CrossRefGoogle Scholar
Zhang, H., Qian, X., Yan, J. Y. and Song, S. H., “Highly efficient invariant-conserving explicit Runge–Kutta schemes for nonlinear Hamiltonian differential equations”, J. Comput. Phys. 418 (2020) 109598; doi:10.1016/j.jcp.2020.109598.CrossRefGoogle Scholar
Zhao, J., Yang, X., Gong, Y., Zhao, X., Yang, X., Li, J. and Wang, Q., “A general strategy for numerical approximations of non-equilibrium models—part I: thermodynamical systems”, Int. J. Numer. Anal. Model. 15 (2018) 884918, available at https://doc.global-sci.org/uploads/Issue/IJNAM/v6n15/615_884.pdf.Google Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

LINEARLY IMPLICIT ENERGY-PRESERVING FOURIER PSEUDOSPECTRAL SCHEMES FOR THE COMPLEX MODIFIED KORTEWEG–DE VRIES EQUATION
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

LINEARLY IMPLICIT ENERGY-PRESERVING FOURIER PSEUDOSPECTRAL SCHEMES FOR THE COMPLEX MODIFIED KORTEWEG–DE VRIES EQUATION
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

LINEARLY IMPLICIT ENERGY-PRESERVING FOURIER PSEUDOSPECTRAL SCHEMES FOR THE COMPLEX MODIFIED KORTEWEG–DE VRIES EQUATION
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *