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.