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Development of a Combined Compact Difference Scheme to Simulate Soliton Collision in a Shallow Water Equation

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Hyperbolic equations and systems Elliptic equations and systems

Published online by Cambridge University Press:  16 March 2016

Ching-Hao Yu  and
Tony Wen-Hann Sheu
Ching-Hao Yu
Affiliation:
Ocean College, Zhejiang University, 866 Yuhangtang Road, Hangzhou, Zhejiang, China
Tony Wen-Hann Sheu*
Affiliation:
Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan Institute of Applied Mathematical Sciences, National Taiwan University, Taipei, Taiwan Center of Advanced Study in Theoretical Sciences (CASTS), National Taiwan University, Taipei, Taiwan
*
*Corresponding author. Email addresses: chyu@zju.edu.cn (C.-H. Yu), twhsheu@ntu.edu.tw (T. W.-H. Sheu)
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Abstract

In this paper a three-step scheme is applied to solve the Camassa-Holm (CH) shallow water equation. The differential order of the CH equation has been reduced in order to facilitate development of numerical scheme in a comparatively smaller grid stencil. Here a three-point seventh-order spatially accurate upwinding combined compact difference (CCD) scheme is proposed to approximate the firstorder derivative term. We conduct modified equation analysis on the CCD scheme and eliminate the leading discretization error terms for accurately predicting unidirectional wave propagation. The Fourier analysis is carried out as well on the proposed numerical scheme to minimize the dispersive error. For preserving Hamiltonians in Camassa- Holm equation, a symplecticity conserving time integrator has been employed. The other main emphasis of the present study is the use of u–P–α formulation to get nondissipative CH solution for peakon-antipeakon and soliton-anticuspon head-on wave collision problems.

Keywords

Camassa-Holm equationseventh-order spatially accurateCCDsymplecticitypeakon-antipeakonsoliton-anticuspon

MSC classification

Secondary: 35L05: Wave equation 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 65M06: Finite difference methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 3 , March 2016 , pp. 603 - 631
Copyright
Copyright © Global-Science Press 2016 

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