The time-fractional convection–diffusion equation (TFCDE) arises in various fields of science and engineering. However, its numerical solution presents challenges stemming from the weak singularity at the initial time. To efficiently address this singularity, temporal discretization is performed via the L2- \(1_{\sigma }\) scheme on a non-uniform graded mesh, and spatial derivatives are approximated by a fourth-order radial basis function compact finite difference (RBF-CFD) method. The linear system from the scheme, a typical tridiagonal system, allows fast solution with the Thomas algorithm, enhancing computational efficiency. The proposed scheme achieves a temporal convergence rate of \({\min \{r\gamma , 2\}}\) and fourth-order spatial convergence, where \(r\) is the mesh grading parameter and \(\gamma \) denotes the order of the Caputo fractional derivative. Rigorous theoretical analyses, including proofs of stability and convergence, are presented. The proposed method is applied to several numerical examples, and the results are compared with those from existing methods in the literature. The results confirm that the method demonstrates good stability and consistently attains the theoretically predicted convergence order.
