An alternating direction implicit method for 2D nonlinear Schrödinger equation with accelerated evaluation of Caputo derivative

We propose an efficient time-space discretization for nonlinear fractional Schrödinger equations involving Caputo tempered derivatives. A new tempered Alikhanov scheme with parameter λ is introduced, together with a fast sum-of-exponentials (SOE) implementation, reducing complexity to O(MKtlogKt) and memory to O(MlogKt). Spatial derivatives are approximated using a compact scheme, and an alternating direction implicit formulation is derived with perturbation terms for stability. A graded time mesh resolves the initial singularity, while adaptive time-stepping ensures long-time efficiency. Stability and maximum-norm error bounds are established via a discrete Grönwall inequality. Numerical tests confirm the theoretical convergence and demonstrate substantial savings in CPU time and storage over classical methods. This work presents a novel nonuniform tempered Alikhanov time-stepping framework for nonlinear tempered fractional Schrödinger equation(NL-TFSEs), combining robustness, high accuracy, and computational scalability.