Solving time-fractional Schrödinger equation via the numerical method on extended space–time sparse grid

In this paper, we focus on the numerical solution of d-dimensional time-fractional Schrödinger equation (TFSE) with nonsmooth potential, and construct the sine pseudospectral/L1-difference scheme on full grid (FG) and space–time sparse grid (STSG). Compared to the conventional STSG, our method introduces a scale factor M, whose optimal value is determined by solution regularity. For the extended STSG framework, we rigorously analyze the degrees of freedom (DOF) and interpolation approximation errors. By integrating the extended STSG with the discretization of Caputo fractional derivative, we establish a fully discrete scheme for solving TFSE. Numerical experiments demonstrate significant computational advantages of the extended STSG method, particularly highlighting enhanced efficiency when employing scale factors M = 2 and M = 4.