Abstract: Optimizing finite difference schemes in spectral space is important for numerical simulations in unsteady flow and aeroacoustics.It is proved that optimized schemes are not always better than high order schemes(in the sense of Taylor expansion).There exists a distance R_cr.Only before the numerical wave traveling this distance,its accumulating error is less than that of a higher order scheme.Beyond this distance,the regularity is reversed.This distance is called here the critic distance of an optimized scheme.It is necessary to control this distance in scheme optimizing.The conventional optimizing objective functions are based on amplification factor revealing the error after one time step,by which it is impossible to specify this distance.In this paper,a new class of objective functions is developed based on accumulating errors,which can control the critic distance directly.

Key words: finite difference method, unsteady flow, computational aeroacoustics, optimal analysis