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Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs

Part of: Infinite-dimensional Hamiltonian systems Equations of mathematical physics and other areas of application Numerical problems in dynamical systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  17 May 2016

Wenjun Cai ,
Huai Zhang  and
Yushun Wang
Wenjun Cai
Affiliation:
Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing 100049, China Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
Huai Zhang
Affiliation:
Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing 100049, China
Yushun Wang*
Affiliation:
Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
*
*Corresponding author. Email addresses:wenjuncai1@gmail.com(W. Cai), hzhang@ucas.ac.cn(H. Zhang), wangyushun@njnu.edu.cn(Y. Wang)
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Abstract

This paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, bandlimited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.

Keywords

Discrete singular convolutiondifferential matrixsymplectic integratorHamiltonian PDEs

MSC classification

Secondary: 35Q55: NLS-like equations (nonlinear Schrödinger) 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 65P10: Hamiltonian systems including symplectic integrators 65M20: Method of lines 65M70: Spectral, collocation and related methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1375 - 1396
Copyright
Copyright © Global-Science Press 2016 

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