An averaged L1 ADI compact difference scheme for the three-dimensional time-fractional mobile/immobile transport equation with weakly singular solutions

In this paper, a three-dimensional (3D) time-fractional mobile/immobile (MIM) transport equation, which incorporates the Caputo time-fractional derivative of order $\alpha \in (0,1)$, is taken into consideration. The space derivatives are discretized using the compact finite difference approximation, and the Caputo time-fractional derivative is estimated by employing the averaged L1 formula. Combining with corresponding alternating direction implicit (ADI) algorithms, the overall computational cost is reduced significantly. Using the discrete energy analysis methodology, we demonstrate that the suggested method possesses temporal second-order convergence and spatial fourth-order convergence under the regularity assumption. Numerical experiments demonstrate that ADI techniques is effective in computing 3D problems.