高精度迎风偏斜格式的比较与分析

A Comparison and Analysis of High Order Upwind-Biased Schemes

利用一种具有任意阶精度的一般显式有限差分公式构造出高精度迎风偏斜格式,并利用Fourier分析法评估了这些迎风偏斜格式的耗散误差与频散误差。结果表明,偶阶精度格式的数值相速度快于实际相速度,而奇阶精度格式的数值相速度慢于实际相速度。并且,偶阶精度格式的耗散误差与频散误差低于相邻的奇阶精度格式。为了检验这些格式的计算性能,在一维问题上进行了应用。首先,考虑恒定风场条件下的一维平流试验。主要选择两种不同的初始条件来评价数值格式的精度,这两种试验问题分别是高斯函数、方波函数。试验结果表明,随着数值格式精度的提高,数值格式的误差逐渐减小。而对于高于六阶精度的格式来说,改进的程度并不是很大。其次,应用各阶格式到具有两种不同初始条件的无粘Burgers方程。数值结果表明,随着数值格式阶数的增加,数值结果也得到了明显改进。而对于高于六阶精度的格式来说,进一步的变化并不明显。总之,在兼顾效率与精度条件下六阶迎风偏斜格式是最好的。

High order upwind-biased schemes based on the general explicit difference formulas with arbitrary order accuracy from Li （2005） are developed. The dissipation and dispersion errors of these upwind-biased schemes are assessed by means of the Fourier analysis. The results show that the numerical phase speeds for even order schemes are faster than the real phase speed. But the numerical phase speeds for odd order schemes are slower than the real phase speed. Moreover, the dissipation and dispersion errors of the even order scheme are smaller than the next lower-order odd scheme. For the upwind-biased schemes derived, there is an increase of the dissipation error in the high wave number range. However, the increase of numerical dissipation in the low wave number range is very small. Some numerical dissipation is needed in the high wave number range for some flow problems. The decision that a numerical modeler should always make is how much numerical dissipation is need. The upwind-biased schemes provide some additional alternatives for adjusting numerical dissipation. These schemes are applied to a onedimensional model in order to test their computational performance. Firstly, consider one-dimensional advection in a constant velocity field. The two types of test problems that are used to evaluate the accuracy of a numerical scheme are the Gaussian function and the square wave function. The first example is the advection of the square wave function. This function reveals a numerical method＇s capability to handle Gibb＇s oscillations that arise in the vicinity of discontinuities. The second example is the advection of the Gaussian function. This function provides a demonstration of each scheme＇s ability to transport well resolved, smoothly varying functions over larger distances. Testing results show that the errors of numerical schemes gradually decrease as the order of the scheme increases and further changes are moderate for higher than sixth order scheme. A high order scheme, although not maintaining a positive definite field, will reduce the problem of negative quantities dramatically. Secondly, consider the inviscid Burgers＇ equation. The inviscid Burgers＇ equation is used as the test model because it is the simplest equation that allows scale collapse （shock formation）, and because it has analytic solutions. There are many important phenomena in the atmosphere that are associated with sharp gradients. One requirement of a computational method for the scale collapse phenomena would seem to be an ability to accurately represent a very sharp gradient in the computational domain. In addition, any numerical dispersion error due to the presence of a shock should not contaminate the smooth solution away from the shock. Numerical results show that there is a drastic improvement when going from the first to sixth order, while further changes are moderate. The results indicate the usefulness of the upwind-biased scheme in handling the advection near discontinuities and the development of scale collapse regions in atmospheric numerical models. In conclusion, the sixth-order upwind-biased scheme appears to be the best balance between efficiency and accuracy. It appears that the sixth-order upwind-biased scheme is well suited for many atmospheric modeling applications where advection plays a significant role.