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

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  12 January 2021

J. L. YAN Open the ORCID record for J. L. YAN [Opens in a new window] ,
L. H. ZHENG Open the ORCID record for L. H. ZHENG [Opens in a new window] ,
L. ZHU Open the ORCID record for L. ZHU [Opens in a new window]  and
F. Q. LU Open the ORCID record for F. Q. LU [Opens in a new window]
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.
*
e-mail: yanjinliang3333@163.com
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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.

Keywords

massenergyinvariant energy quadratization methodFourier pseudospectral methodcomplex modified Korteweg–de Vries equation

MSC classification

Primary: 65M22: Solution of discretized equations
Secondary: 65M20: Method of lines 65M70: Spectral, collocation and related methods 65L05: Initial value problems
Type
Research Article
Information
The ANZIAM Journal , Volume 62 , Issue 3 , July 2020 , pp. 256 - 273
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Copyright
© Australian Mathematical Society 2021

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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) 166–185; 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) 535–549; 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) 193–204, 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) 915–919; 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) 224–231; 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) B135–B156; 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) B138–B167; 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) 223–228; 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) 86–95; 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) 520–531; 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) 749–759; 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) 940–952; 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) 1516–1523; 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), 275–285; 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) 503–514; 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) 566–578; 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) 515–525; 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) 104–127; 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) 884–918, available at https://doc.global-sci.org/uploads/Issue/IJNAM/v6n15/615_884.pdf.Google Scholar