This paper proposes a new approach for power system transient stability simulation, which is based on a semi-analytical solution (SAS) of power system differential-algebraic equations. In this paper, an SAS is derived using the Adomian decomposition method as a closed-form explicit function of symbolic variables such as time, the initial state, and other variables on system conditions and; hence, it can directly give a power system's dynamic trajectory being accurate for a certain time window. Unlike a traditional numerical integration based simulation approach, the proposed new approach offline derives an SAS and online evaluates the SAS by plugging in values of symbolic variables for a series of time windows making up the desired simulation period. This paper further studies the maximum length of the time window for an SAS being accurate and proposes a divergence indicator for simulation using adaptive time windows. Implementation of this new approach on parallel computers is also studied. The new approach is validated through contingency simulation of the IEEE 10-generator 39-bus system with detailed generator models.