  A rotation-two-component Camassa-Holm (R2CH) system was proposed recently to
describe the motion of shallow water waves under the influence of gravity. This
is a highly nonlinear and strongly coupled system of partial differential
equations. A crucial issue in designing numerical schemes is to preserve
invariants as many as possible at the discrete level. In this paper, we present
a provable implicit nonlinear difference scheme which preserves at least three
discrete conservation invariants: energy, mass, and momentum, and prove the
existence of the difference solution via the Browder theorem. The error
analysis is based on novel and refined estimates of the bilinear operator in
the difference scheme. By skillfully using the energy method, we prove that the
difference scheme not only converges unconditionally when the rotational
parameter diminishes, but also converges without any step-ratio restriction for
the small energy case when the rotational parameter is nonzero. The convergence
orders 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 spatial
stepsize, respectively. The theoretical predictions are confirmed by a properly
designed two-level iteration scheme. Comparing with existing numerical methods
in the literature, the proposed method demonstrates its effectiveness for
long-time simulation over larger domains and superior resolution for both
smooth and non-smooth initial values.
