最优差分方案

在数值预报和数值模拟中,描述空间微分项的最主要的方法是有限差分法,但使用差分方法会引入截断误差.伍荣生1979年指出,通过在原物理场的基础上构造一个新的物理场,替代原物理场进行差分计算,可以达到减小误差的目的.该文是伍荣生1979年工作的继续,目的在于解释伍荣生1979年所构造的差分格式并得到更为一般化的差分格式.文中给出新的差分格式结合了经典有限差分方法的快速计算和谱方法的高精度的优点.如果在一个给定的网格上对气象要素场进行离散傅里叶级数展开,则基函数(正弦或余弦)的频谱是事免已知的.作者将伍荣生1979年构造物理场的方法视为对物理场的一次平滑,探讨了获取二次平滑场、多次平滑的一般化方法.获取平滑场的基奉原理是使得在固定频谱上的差分逼近程度达到最优.通过对频谱上的累计误差的下降速度分析表明,平滑次数的上限为3次.数值分析的结果表明,二次平滑的最大误差是未作任何平滑的最大误差的0.04倍,在使用相同计算代价的情况下,二次平滑的最大误差是经典的差分格式的0.3倍.平流试验的结果也表明,新的差分格式即一次平滑、二次平滑方案的结果远远优于经典的差分格式.新的差分格式意义在于,在不加密网格的情况下提供了一条提高数值计算精度的途径.

An optimum scheme for finite difference

In numerical prediction and numerical modeling, the general method to describe differential term in space is finite difference method, however, the using of finite difference method will introduce truncation error. Wu (1979) proposed that in order to improve the accuracy of difference term, a new field was constructed to replace the original physical field in the difference term. This paper is a sister paper of Wu (1979), the main purpose is to interpret the value of Wu (1979), and furthermore to give some more general difference themes. The difference theme in this paper combines both the advantages of finite difference method (fast calculating) and the spectral method (high accuracy). If a discrete Fourier expansion is made on a given grid, the frequency spectrum of the base function (sine or cosine) is fixed. In this paper, the generalized method of finding a 2-order (or more times) smoothing field is explored. The fundamental philosophy to obtain the smoothing field is making an optimum approximation at the fixed frequency spectrum. The upper threshold of smoothing was determined as 3 through observing the decreasing speed of the cumulative error of the frequency spectrum. The results of the numerical analysis reveal that the maximum error of the 2-order smoothing scheme is 0.04 of the classical scheme without any smoothing and 0. 3 of the classical scheme with the same computation cost. The advection experiment also suggests that the new scheme is far more excellent than the classical scheme. The new difference scheme supplies a new road which improves the accuracy of numerical calculating without adding the grids.