Abstract In this study, we introduce an alternating direction implicit (ADI) compact finite‐difference scheme for a time‐fractional convection‐diffusion equation (TFCDE) governing groundwater pollution. The considered TFCDE has a weak singularity near the initial time . To address this singularity, we employ the ‐approximation of the Liouville–Caputo time‐fractional derivative on a graded temporal mesh. In the space direction, we define compact operators to discretize the space derivatives and then employ the ADI method to reduce the computational complexity of the method. Moreover, the unconditional stability and convergence of the proposed method are established using energy method. The proposed method has a temporal accuracy of order , where denotes the grading parameter and attains fourth‐order accuracy in space. Numerical examples are provided to demonstrate the accuracy and effectiveness of the proposed method, with two cases having direct physical relevance. We compute the pollutant concentration near the source and analyze how the order of the time‐fractional derivative affects the diffusion and convection processes. It is shown that the proposed graded mesh technique possesses an advantage over the uniform mesh method and the fractional model provides better numerical results than the classical model.
Roul et al. (Thu,) studied this question.