一类新的差分格式优化目标函数

A NEW CLASS OF OBJECTIVE FUNCTIONS FOR FINITE DIFFERENCE SCHEME OPTIMIZING

对差分格式进行优化处理可以提高其谱精度。与高精度（指Ｔａｙｌｏｒ展开精度）格式相比，优化格式放大因子的误差随波数的变化不是单调的，而是必然会出现极值点，这样就存在临界距离Ｒｃｒ，在此距离内优化格式描述的数值波的积累误差小于高精度格式，而超出此距离后优化格式的误差反而大，对于非定常流及气动声学计算来说，控制差分格式的临界距离是必要的。一般的优化目标函数以每个时间推进步的误差为基础（即放大因子法），Ｒｃｒ不能在优化过程中确定。对此进行分析，指出积累误差的重要性并提出以此为基础的新的优化目标函数，这样在对格式进行优化时可以直接指定临界距离，从而为控制谱精度提供方便。

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.