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Geometric Numerical Integration for Peakon b-Family Equations

Part of: Numerical problems in dynamical systems Hyperbolic equations and systems Waves Partial differential equations, boundary value problems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  15 January 2016

Wenjun Cai ,
Yajuan Sun  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
Yajuan Sun*
Affiliation:
LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, 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), sunyj@lsec.cc.ac.cn (Y. Sun), wangyushun@njnu.edu.cn (Y. Wang)
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Abstract

In this paper, we study the Camassa-Holm equation and the Degasperis-Procesi equation. The two equations are in the family of integrable peakon equations, and both have very rich geometric properties. Based on these geometric structures, we construct the geometric numerical integrators for simulating their soliton solutions. The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties, however they also have the significant difference, for example there exist the shock wave solutions for the Degasperis-Procesi equation. By using the symplectic Fourier pseudo-spectral integrator, we simulate the peakon solutions of the two equations. To illustrate the smooth solitons and shock wave solutions of the DP equation, we use the splitting technique and combine the composition methods. In the numerical experiments, comparisons of these two kinds of methods are presented in terms of accuracy, computational cost and invariants preservation.

Keywords

Symplectic integratorsplitting methodWENO schememultisymplectic integratorpeakonshockpeakon

MSC classification

Secondary: 35L65: Conservation laws 65M70: Spectral, collocation and related methods 65N06: Finite difference methods 65P10: Hamiltonian systems including symplectic integrators 74J40: Shocks and related discontinuities
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 1 , January 2016 , pp. 24 - 52
Copyright
Copyright © Global-Science Press 2016 

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