This paper explores an alternative, semi-analytical approach to solution of the initial value problem of differentialalgebraic equations modeling a power system. Different from the traditional numerical integration based approach, this new approach applies the Adomian Decomposition Method to derive an approximate solution, called a semi-analytic solution (SAS), as a closed-form explicit function of time, the initial state and parameters on the system condition. Such a solution directly gives the power system's dynamic trajectory starting from an initial state that is accurate over a certain time window. Then, a multi-stage scheme evaluating the same SAS repeatedly for sequential time windows is able to give the system's trajectory for a desired simulation period without iterative computations as numerical integration does. The new approach is tested for power system transient stability simulation on a 3-machine, 9-bus power system and the IEEE 10-machine, 39-bus system, and its accuracy and time performance are compared with the 4th order Runge-Kutta method.