This paper presents a numerical scheme for approximating the solution of a three-dimensional time-fractional advection–diffusion equation (TFADE) that exhibits a weak singularity at \(t=0\) . The temporal fractional derivative is discretized using the \(L2-1_\sigma \) scheme on non-uniform meshes, while the space derivative is discretized using a fourth-order compact finite difference (CFD) scheme. The resultant fully discrete scheme is computationally expensive. We propose an alternating direction implicit (ADI) scheme to reduce the computational complexity of the method. A theoretical analysis of the stability and convergence of the proposed numerical method is carried out using the \(H^1\) -norm and \(L^2\) -norm. The method’s performance, robustness and accuracy are tested through numerical experiments. Additionally, a comparison of numerical results obtained on uniform, graded and variable graded meshes is provided to highlight the advantages of the proposed non-uniform mesh method over the uniform mesh method. We compare the numerical results obtained by the present method and the methods reported in Zhou et al. (Numer Algorithms 96:1533–1551, 2024) and Roul and Rohil (Comput Math Appl 126:1–13, 2022). The numerical results presented both in tabular and graphical forms confirm the scheme’s high accuracy and versatility.
