CENTRAL DIFFERENCE SCHEMES WITH HIGH ORDER ACCURACY ON NONUNIFORM GRIDS: THEORETICAL DERIVATION AND IDEAL TEST
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摘要: 传统的高阶精度有限差分格式通常是在均匀网格的基础上推导得到的，在非均匀网格的情况下它会出现精度退化的问题。基于泰勒展开方法构造了一种适用于非均匀网格的2阶、4阶和6阶精度中央有限差分方案，利用Burgers方程和一维平流方程对新方案的性能进行测试，着重分析新方案对其误差大小及分布形态的改进效果。数值模拟结果表明：在非均匀网格下，提高差分方案的精度可明显减小数值解误差(降低了70%~88%)，特别是当差分精度从2阶提高到4阶的时候。同时，高阶精度方案在梯度变化较大或者网格距较粗区域的模拟结果更有优势，4阶和6阶精度方案在以上区域的误差远小于2阶精度方案。方案可用于提高数值天气预报模式中非均匀分层模式的垂直差分计算精度。Abstract: Traditional central finite difference schemes are derived on uniform grids. However, the precision of such schemes will decrease on non-uniform grids. This article derives the 2nd, 4th, and 6th order central finite difference schemes using Taylor expansion method. Burgers equation and 1-D advection equation are numerically solved by the new schemes on non-uniform grids. Comparing the numerical solution errors, we find that higher accuracy schemes can reduce up to 70%~88% numerical errors, especially when the accuracy is improved from the 2nd order to the 6th order. It is also found that higher accuracy schemes have advantages in regions of steep gradient or coarser resolution, in which the errors of 4th and 6th order schemes are much smaller than that of the 2nd order scheme.
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表 1 2阶、4阶和6阶非均匀网格差分方案的数值解的误差L2
误差L2 2阶 4阶 6阶 1.5 s 6.57×10-4 2.65×10-5 2.54×10-5 2.0 s 4.15×10-4 5.91×10-5 5.79×10-5 2.5 s 3.31×10-4 9.90×10-5 1.00×10-4 
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表 2 2阶、4阶和6阶非均匀网格差分方案的平流方程数值解的误差L2
误差L2 2阶 4阶 6阶 1 s 2.52×10-3 2.97×10-4 2.96×10-4 3 s 8.86×10-3 3.26×10-3 1.55×10-3 5 s 3.55×10-2 1.58×10-2 9.51×10-3 10 s 8.43×10-2 4.67×10-2 3.39×10-2
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