Finite-Volume-Particle Methods for the Two-Component Camassa-Holm System

We study the two-component Camassa-Holm (2CH) equations as a model for the long time water wave propagation. Compared with the classical Saint-Venant system, it has the advantage of preserving the waves amplitude and shape for a long time. We present two different numerical methods—finite volume (FV) and hybrid finite-volume-particle (FVP) ones. In the FV setup, we rewrite the 2CH equations in a conservative form and numerically solve it by the central-upwind scheme, while in the FVP method, we apply the central-upwind scheme to the density equation only while solving the momentum and velocity equations by a deterministic particle method. Numerical examples are shown to verify the accuracy of both FV and FVP methods. The obtained results demonstrate that the FVP method outperforms the FV method and achieves a superior resolution thanks to a low-diffusive nature of a particle approximation.