We develop an efficient numerical scheme for solving variable-order time-fractional mobile/immobile transport problems with discontinuous diffusion coefficients. The proposed method combines an \(L1\) approximation for the Caputo fractional derivative in time with a symmetric interior penalty discontinuous Galerkin discretization in space, specifically designed to handle solution non-smoothness caused by discontinuous diffusion coefficients across interfaces Γ. To resolve the solution’s initial singularity, we implement a graded temporal mesh that concentrates grid points near the initial time. A body-fitted mesh partition for spatial domain is applied and curved elements with a curved edge on the curved interface are considered. Through the rigorous analysis, we establish the unconditional stability of the fully discrete scheme and prove optimal convergence rate of \(O\left(N^{-\min\{1,r\delta\}}+h^{\min\{k+1,s\}}\right)\) in the \(l^{\infty}(0,T;L_2)\) norm. Some numerical experiments confirm these theoretical results and demonstrate the method’s robustness in handling interface problems with variable fractional orders.
