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On Invariant-Preserving Finite Difference Schemes for the Camassa-Holm Equation and the Two-Component Camassa-Holm System

Part of: Equations of mathematical physics and other areas of application Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  12 April 2016

Hailiang Liu  and
Terrance Pendleton
Hailiang Liu*
Affiliation:
Iowa State University, Mathematics Department, Ames, IA 50011, USA
Terrance Pendleton
Affiliation:
Iowa State University, Mathematics Department, Ames, IA 50011, USA
*
*Corresponding author. Email addresses:hliu@iastate.edu (H. Liu), tlpendle@iastate.edu (T. Pendleton)
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Abstract

The purpose of this paper is to develop and test novel invariant-preserving finite difference schemes for both the Camassa-Holm (CH) equation and one of its 2-component generalizations (2CH). The considered PDEs are strongly nonlinear, admitting soliton-like peakon solutions which are characterized by a slope discontinuity at the peak in the wave shape, and therefore suitable for modeling both short wave breaking and long wave propagation phenomena. The proposed numerical schemes are shown to preserve two invariants, momentum and energy, hence numerically producing wave solutions with smaller phase error over a long time period than those generated by other conventional methods. We first apply the scheme to the CH equation and showcase the merits of considering such a scheme under a wide class of initial data. We then generalize this scheme to the 2CH equation and test this scheme under several types of initial data.

Keywords

Camassa-Holm equationPeakon solutionsenergy-preserving methods

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M12: Stability and convergence of numerical methods 35Q53: KdV-like equations (Korteweg-de Vries)
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 4 , April 2016 , pp. 1015 - 1041
Copyright
Copyright © Global-Science Press 2016 

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