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. The 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 convergence is proved by the technical energy argument and induction method with the optimal convergence order \(\mathcal {O}(\tau ^2+h^4)\) in the sense of the maximum norm. The stability under mild conditions can be achieved based on the uniform boundedness of the numerical solution. The present scheme is very efficient in practical computation since only a linear system needs to be solved at each time. The extensive numerical examples verify our theoretical results and demonstrate the scheme’s superiority when compared with state-of-the-art those in the references.

在本文中，一个新开发的三点四阶紧致算子被用来为本杰明-波纳-马奥尼-伯格斯（BBMB）方程构造一个有效的紧致有限差分方案。详细的推导是基于降阶方法和三级线性化技术进行的。详细研究了保守不变性，有界性和独特的可解性。通过技术能论证和归纳法证明收敛性为最优收敛阶\（\ mathcal {O}（\ tau ^ 2 + h ^ 4）\）在最大规范的意义上。基于数值解的均匀有界性，可以实现在温和条件下的稳定性。由于每次仅需要求解线性系统，因此本方案在实际计算中非常有效。大量的数值示例验证了我们的理论结果，并证明了该方案与参考文献中的最新技术相比具有优越性。