A numerical approach for solving the time-Fractional Mobile-Immobile Transport Equation
Fractional diffusion models provide a powerful framework for describing anomalous transport phenomena in heterogeneous porous media. The Mobile-Immobile model is a fundamental approach for characterizing such anomalous diffusion, specifically addressing the delayed solute transport caused by mass transfer between mobile and immobile regions. In this study, we develop and analyze two fully discrete numerical schemes for the one-dimensional time-fractional Mobile-Immobile model governed by the Caputo derivative of order $\alpha\in (0,1)$. The spatial discretization is carried out using a non-symmetric interior penalty discontinuous Galerkin method, while the temporal derivative is approximated by the Crank-Nicolson L1 and L2-$1_\sigma$ formulas, resulting in two distinct schemes with high-order accuracy. Rigorous stability and error analyses are established, showing that the methods achieve optimal convergence rates depending on the regularity of the exact solution. Numerical experiments verify the theoretical predictions and demonstrate the efficiency and robustness of the proposed schemes. The presented framework provides a reliable and accurate approach for simulating complex fractional transport processes in porous media and related fields.