  In the paper, a newly developed three-point fourth-order compact operator is
utilized to construct an efficient compact finite difference scheme for the
Benjamin-Bona-Mahony-Burgers' (BBMB) equation. Detailed derivation is carried
out based on the reduction order method together with a three-level linearized
technique. The conservative invariant, boundedness and unique solvability are
studied at length. The uniform convergence is proved by the technical energy
argument with the optimal convergence order $\mathcal{O}(\tau^2+h^4)$ in the
sense of the maximum norm. The almost unconditional stability can be achieved
based on the uniform boundedness of the numerical solution. The present scheme
is very efficient in practical computation since only a system of linear
equations with a symmetric circulant matrix needing to be solved at each time
step. The extensive numerical examples verify our theoretical results and
demonstrate the superiority of the scheme when compared with state-of-the-art
those in the references.
