Numerical solutions to a two-component Camassa-Holm equation

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.