In this paper, some algorithmic and computational advances are presented for power system dynamic simulations. The heart is a Schur-complement-based solution algorithm, stemming from domain decomposition methods, applied to the differential-algebraic equation model. This algorithm is then accelerated computationally, by employing parallel computing techniques, and numerically, by exploiting time-scale decomposition and localization. Models of a real medium-scale system and a realistic large-scale test system are used for the performance evaluation of the proposed methods.