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

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

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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