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A rotation-two-component Camassa-Holm (R2CH) system was proposed recently todescribe the motion of shallow water waves under the influence of gravity. Thisis a highly nonlinear and strongly coupled system of partial differentialequations. A crucial issue in designing numerical schemes is to preserveinvariants as many as possible at the discrete level. In this paper, we presenta provable implicit nonlinear difference scheme which preserves at least threediscrete conservation invariants: energy, mass, and momentum, and prove theexistence of the difference solution via the Browder theorem. The erroranalysis is based on novel and refined estimates of the bilinear operator inthe difference scheme. By skillfully using the energy method, we prove that thedifference scheme not only converges unconditionally when the rotationalparameter diminishes, but also converges without any step-ratio restriction forthe small energy case when the rotational parameter is nonzero. The convergenceorders in both settings (zero or nonzero rotation parameter) are $O(\tau^2 +h^2)$ for the velocity in the $L^\infty$-norm and the surface elevation in the$L^2$-norm, where $\tau$ denotes the temporal stepsize and $h$ the spatialstepsize, respectively. The theoretical predictions are confirmed by a properlydesigned two-level iteration scheme. Comparing with existing numerical methodsin the literature, the proposed method demonstrates its effectiveness forlong-time simulation over larger domains and superior resolution for bothsmooth and non-smooth initial values.

Error estimates of invariant-preserving difference schemes for the rotation-two-component Camassa--Holm system with small energy