On Invariant-Preserving Finite Difference Schemes for the Camassa-Holm Equation and the Two-Component Camassa-Holm System (Communications in Computational Physics, 2018)
The purpose of this paper is to develop and test novel invariant-preservingfinite difference schemes for both the Camassa-Holm (CH) equation and one of its2-component generalizations (2CH). The considered PDEs are strongly nonlinear, admittingsoliton-like peakon solutions which are characterized by a slope discontinuityat the peak in the wave shape, and therefore suitable for modeling both short wavebreaking and long wave propagation phenomena. The proposed numerical schemesare shown to preserve two invariants, momentum and energy, hence numerically producingwave solutions with smaller phase error over a long time period than thosegenerated by other conventional methods. We first apply the scheme to the CH equationand showcase the merits of considering such a scheme under a wide class of initialdata. We then generalize this scheme to the 2CH equation and test this scheme underseveral types of initial data.