This paper is concerned with a numerical method for the time-fractional Benjamin–Bona–Mahony (BBM) equation whose solution typically exhibits a weak singularity at the initial time. Lyu and Vong (2019) presented a linearized difference method of second-order in space and third-order in time. We improve their result by proposing a linearized and compact difference method which is fourth-order in space while keeping third-order in time. By using discrete energy analysis, the unconditional convergence of the proposed method is rigorously proved and the optimal $H^1$-norm error estimate is obtained. Numerical results confirm the theoretical convergence result.

本文涉及时间分数阶本杰明-波纳-马洪尼（BBM）方程的数值方法，其解在初始时通常表现出弱的奇异性。Lyu和Vong（2019）提出了空间二阶和时间三阶的线性差分方法。我们通过提出一种线性且紧凑的差分方法来提高其结果，该方法在空间上保持四阶同时在空间上是四阶的。通过离散能量分析，严格证明了该方法的无条件收敛性和最优性。$H^{\mathrm{1个}}$-获得标准误差估计。数值结果证实了理论收敛结果。