A sixth-order dual preserving algorithm for the Camassa-Holm equation
Resource
Journal of Computational and Applied Mathematics, 233(11), 2767-2778
Journal
Journal of Computational and Applied Mathematics
Journal Volume
233
Journal Issue
11
Pages
2767-2778
Date Issued
2010
Date
2010
Author(s)
Abstract
The paper presents a sixth-order numerical algorithm for studying the completely integrable Camassa-Holm (CH) equation. The proposed sixth-order accurate method preserves both the dispersion relation and the Hamiltonians of the CH equation. The CH equation in this study is written as an evolution equation, involving only the first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step method that first solves the evolution equation by a sixth-order symplectic Runge-Kutta method and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The sixth-order symplectic Runge-Kutta time integrator is preferable for an equation that possesses a Hamiltonian structure. We illustrate the ability of the proposed scheme by examining examples involving peakon or peakon-like solutions. We compare the computed solutions with exact solutions or asymptotic predictions. We also demonstrate the ability of the symplectic time integrator to preserve the Hamiltonians. Finally, via a smooth travelling wave problem, we compare the accuracy, elapsed computing time, and rate of convergence among the proposed method, a second-order two-step algorithm, and a completely integrable particle method. © 2009 Elsevier B.V. All rights reserved.
Type
journal article
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