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An Energy-Preserving Wavelet Collocation Method for General Multi-Symplectic Formulations of Hamiltonian PDEs

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

Published online by Cambridge University Press:  02 November 2016

Yuezheng Gong  and
Yushun Wang
Yuezheng Gong*
Affiliation:
Jiangsu Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
Yushun Wang*
Affiliation:
Jiangsu Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
*
*Corresponding author. Email addresses:gyz8814@163.com (Y. Gong), wangyushun@njnu.edu.cn (Y.Wang)
*Corresponding author. Email addresses:gyz8814@163.com (Y. Gong), wangyushun@njnu.edu.cn (Y.Wang)
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Abstract

In this paper, we develop a novel energy-preserving wavelet collocation method for solving general multi-symplectic formulations of Hamiltonian PDEs. Based on the autocorrelation functions of Daubechies compactly supported scaling functions, the wavelet collocation method is conducted for spatial discretization. The obtained semi-discrete system is shown to be a finite-dimensional Hamiltonian system, which has an energy conservation law. Then, the average vector field method is used for time integration, which leads to an energy-preserving method for multi-symplectic Hamiltonian PDEs. The proposed method is illustrated by the nonlinear Schrödinger equation and the Camassa-Holm equation. Since differentiation matrix obtained by the wavelet collocation method is a cyclic matrix, we can apply Fast Fourier transform to solve equations in numerical calculation. Numerical experiments show the high accuracy, effectiveness and conservation properties of the proposed method.

Keywords

Energy-preservingaverage vector field methodwavelet collocation methodnonlinear Schrödinger equationCamassa-Holm equation

MSC classification

Secondary: 65M06: Finite difference methods 65M70: Spectral, collocation and related methods 65T50: Discrete and fast Fourier transforms 65Z05: Applications to physics
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016 , pp. 1313 - 1339
Copyright
Copyright © Global-Science Press 2016 

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