In this paper the space variable-order fractional Schr ¨odinger equation
(VOFSE) is studied numerically, where the variable-order fractional derivative is
described here in the sense of the quantum Riesz-Feller definition. The proposed
numerical method is the weighted average non-standard finite difference method
(WANSFDM). Special attention is given to study the stability analysis and the convergence
of the proposed method. Finally, two numerical examples are provided to
show that this method is reliable and computationally efficient.