Schubert determinantal ideals are an important class of polynomial ideals that have attracted significant attention in algebraic combinatorics, representation theory, and computational algebra in the last two decades. The Schubert determinantal ideal of a permutation is naturally associated with the matrix Schubert variety and is closely related to the double Schubert polynomial of that permutation. In particular, with Fulton generators identified as Grobner bases of Schubert determinantal ideals w.r.t. any anti-diagonal term order, minimal Grobner bases for such ideals were also studied, where the authors introduced the notion of elusive minors and proved that they form minimal Grobner bases of Schubert determinantal ideals. Furthermore, for Schubert determinantal ideals, while all the elusive minors form the reduced Grobner bases when the defining permutations are vexillary, in the non-vexillary case we derived an explicit formula for computing the reduced Grobner basis from elusive minors which avoids all algebraic operations. Based on this explicit formula, we developed an algorithm named RedGBSchubert to compute the reduced Grobner bases of Schubert determinantal ideals, where this new algorithm outperforms the built-in function InterReduce in Maple for computing the reduced grobner bases for complex Schubert determinantal ideals in terms of computational efficiency.