Linearly Implicit Invariant-Preserving Decoupled Difference Scheme For The Rotation-Two-Component Camassa--Holm System

In this paper, we develop, analyze and numerically test an invariant-preserving three-level linearized implicit difference scheme for a rotation-two-component Camassa--Holm system [L. Fan, H. Gao, and Y. Liu, Adv. Math. 291 (2016), pp. 59--89], which contains strongly nonlinear terms and high-order derivative terms. We prove that the numerical scheme is uniquely solvable and second-order convergent for both the spatial and temporal discretizations. Optimal error estimates for the velocity in the $L^{\infty}$-norm and for the surface elevation in the $L^2$-norm are obtained under a suitable step-size ratio restriction based on energy analysis. In particular, the analysis for the strongly nonlinear term is carried out by splitting it into several recursive expressions. Moreover, the numerical scheme preserves at least two conservation invariants: mass and energy. Extensive numerical experiments including long time simulations for zero$/$nonzero rotation parameters demonstrate solution behavior and verify the convergence results as well as mass/energy conservation.