Time-domain simulation is an important method for researching the transient stability of a power system. This approach mainly involves numerical solution of Differential-Algebraic Equations (DAE). Currently, the most commonly used simulation method first uses an implicit trapezoidal rule to discretize ordinary differential equations, and then employs a Newton-like method to solve the system of nonlinear equations. For a large-scale power system, such a simulation method is time consuming because multiple iterations are usually required in the integration of each time step. This paper proposes a numerical method based on the Kahan method to realize fast transient stability simulation, wherein the electro-mechanical transient behavior of a power system is described by a quadratic differential-algebraic system, and then the Kahan method is used to solve the nonlinear system of DAEs. Based on the Kahan method, the proposed simulation method is only linearly implicit, and the integration at each time step can be computed by solving a single linear system. Thus it has better computational efficiency than existing methods. Case studies conducted on a 5-generator 18-bus and IEEE 145-bus test systems have confirmed the effectiveness of the proposed method. This work is supported by the Science and Technology Project of State Grid Corporation of China (No. XTB17201800225).