This paper proposes a novel method for power system dynamic simulation that solves power system differential algebraic equations by a semi-analytical and semi-numerical approach using continued fractions. The method implements a two-stage scheme to enhance online performance of simulation: the offline derivation stage finds approximate analytical solutions, so-called “semi-analytical solutions,”for state variables of dynamic devices, such as generators in the form of power series of time with symbolic coefficients about system conditions; the online evaluation stage substitutes values on actual system conditions for symbolic coefficients, then transforms the solution into a continued fraction to prolong its time interval of accuracy, and finally calculates the system's trajectory over consecutive, adaptive time intervals for expected simulation results. A priori error bound for continued fractions is proposed to enable the simulation on adaptive time intervals. Compared with the conventional numerical simulation methods, the proposed continued fraction-based method has a fast simulation speed and a good suitability for parallel computing. The method is demonstrated and tested on the IEEE 9-bus system, the IEEE 39-bus system, and Polish 327-generator 2383-bus system.