In this paper, we study the numerical method for time-fractional KdV–Burgers’ equation with initial singularity. The famous $L2$-$1_{\sigma }$ formula on graded meshes is adopted to approximate the Caputo derivative. Meanwhile, a nonlinear finite difference method on uniform grids is deduced for spatial discretization. The proposed method is second order in time and first order in space. With the help of the fractional Grönwall inequality, the unconditional stability and convergence of the current scheme are analyzed based on some skills. To raise the accuracy in spatial direction, a second order method is then carefully deduced. At last, theoretical results are verified by numerical experiments.

在本文中，我们研究了具有初始奇异性的时间分数阶KdV-Burgers方程的数值方法。著名$\mathrm{大号}2$--${\mathrm{1个}}_{\sigma }$采用渐变网格上的公式近似Caputo导数。同时，推导了均匀网格上的非线性有限差分法进行空间离散化。所提出的方法是时间上的二阶和空间上的一阶。借助分数Grönwall不等式，基于一些技巧对当前方案的无条件稳定性和收敛性进行了分析。为了提高空间方向的精度，然后仔细推导了二阶方法。最后，通过数值实验验证了理论结果。