Considering the initial singularity in the solution, we develop an accelerated tempered Alikhanov formula with a tempering parameter \(\lambda \) on nonuniform time grids. We study the maximum principle and energy dissipation law for the continuous tempered fractional Allen-Cahn problem. Based on this analysis, we propose a nonuniform tempered Alikhanov time-stepping scheme for the Allen-Cahn problem, in which a sum-of-exponentials (SOE) algorithm is employed to efficiently approximate the tempered Caputo fractional derivative. Spatial derivatives are discretized using a fourth-order compact finite-difference operator and efficiently implemented through a fast discrete sine transform (DST) via the FFT algorithm. The scheme satisfies a discrete maximum property and leveraging the convolution framework of the consistency error, we provide precise error estimates in maximum-norm that accurately capture the temporal smoothness of the solution. We establish a tempered analogue of the fractional Grönwall inequality and demonstrate its applicability by deriving representative convergence estimates. A dynamic time-stepping technique is also introduced to enhance long-time simulation efficiency, achieving a computational cost of \(\mathcal {O}(MK_t \log M \log K_t)\) and memory usage of \(\mathcal {O}(M \log K_t)\) , where M and \(K_t\) are the spatial and temporal grid sizes, respectively. Furthermore, an accelerated compact ADI scheme is constructed. Numerical results confirm the accuracy of the analysis and the efficiency of the proposed algorithm, marking it as the first accelerated nonuniform tempered Alikhanov compact scheme that maintains the maximum principle for the Allen–Cahn problem.
