Frequency response optimized integrators considering second order derivative are proposed in this paper. Based on the proposed numerical integrators, and others which also consider second order derivative, this paper puts forward a novel power system transient simulation scheme. Instead of using a unique numerical integrator, the proposed simulation scheme chooses proper ones according to the dominant frequency component of the differential state variables. With the proposed simulation scheme, computational efficiency is improved by using large step sizes without sacrificing accuracy. Numerical case studies demonstrate the validity and efficiency of the simulation scheme.