Numerical solutions to a two-component Camassa-Holm equation

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:336

Article

Yu, Ching-Hao¹  Feng, Bao-Feng²  Sheu, Tony W. H.^3,4,5 

[1] Sichuan Univ, State Key Lab Hydraul & Mt River Engn, Chengdu 610000, Sichuan, Peoples R China

[2] Univ Texas Rio Grande Valley, Sch Math & Stat Sci, Edinburg, TX 78539 USA

[3] Natl Taiwan Univ, Dept Engn Sci & Ocean Engn, 1,Sec 4,Roosevelt Rd, Taipei, Taiwan

[4] Natl Taiwan Univ, Inst Math & Appl Math, Taipei, Taiwan

[5] Natl Taiwan Univ, CASTS, Taipei, Taiwan

关键词: Two-component Camassa-Holm equation;    Inhomogeneous Helmholtz equation;    Combined compact difference scheme;    peakon-antipeakon;    Hamiltonians;    Casimir function;   

DOI  :  10.1016/j.cam.2017.12.043

来源: Elsevier

【 摘 要 】

In the present paper, a three-step iterative algorithm for solving a two-component Camassa Holm (2CH) equation is presented. In the first step, the time-dependent equation for the horizontal fluid velocity with nonlinear convection is solved. Then an inhomogeneous Helmholtz equation is solved. Finally, the equation for modeling the transport of density is solved in the third step. The differential order of 2CH equation has been reduced in order to facilitate numerical scheme development in a comparatively smaller grid stencil. In this study, a fifth-order spatially accurate upwinding combined compact difference scheme (UCCD5) which differs from that in Sheu et al. (2011) is developed in a four-point grid stencil for approximating the first-order derivative term. For the purpose of retaining long-time Hamiltonians in the 2CH equation, the time integrator (or time-stepping scheme) chosen is symplectic. Various numerical experiments such as the single peakon, peakon antipeakon interaction and dam-break problems are conducted to illustrate the effectiveness of the proposed numerical method. It is shown that both the Hamiltonians and Casimir functions are conserved well for all problems. (C) 2018 Elsevier B.V. All rights reserved.

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