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In the paper, a newly developed three-point fourth-order compact operator isutilized to construct an efficient compact finite difference scheme for theBenjamin-Bona-Mahony-Burgers' (BBMB) equation. Detailed derivation is carriedout based on the reduction order method together with a three-level linearizedtechnique. The conservative invariant, boundedness and unique solvability arestudied at length. The uniform convergence is proved by the technical energyargument with the optimal convergence order $\mathcal{O}(\tau^2+h^4)$ in thesense of the maximum norm. The almost unconditional stability can be achievedbased on the uniform boundedness of the numerical solution. The present schemeis very efficient in practical computation since only a system of linearequations with a symmetric circulant matrix needing to be solved at each timestep. The extensive numerical examples verify our theoretical results anddemonstrate the superiority of the scheme when compared with state-of-the-artthose in the references.

Uniform convergence and stability of linearized fourth-order conservative compact scheme for Benjamin-Bona-Mahony-Burgers' equation