Potential disagreement in the result induced by discontinuities is revealed in this paper between a novel power system transient simulation scheme using numerical integrators considering second order derivative and conventional ones using numerical integrators considering first order derivative. The disagreement is due to the formula of the different numerical integrators. An improved method for dealing with discontinuities in the novel transient simulation scheme is proposed to resolve the disagreement. The effectiveness of the improved method is demonstrated and verified via numerical case studies. Although the disagreement is studied on and the improved method is proposed for a particular transient simulation scheme, similar conclusions also apply to other ones using numerical integrators considering high order derivative.