Nonstaggered Grid under High-Order Finite-Difference Scheme—Analysis Based on a Shallow Water Equation
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摘要: 非跳点网格在模式动力—物理过程的耦合方面具有独特的优势，但是由于二阶精度差分方案下非跳点网格频散误差较大而很少被使用于数值天气预报模式。随着近年来数值模式计算精度的不断提高，非跳点网格在频散关系方面的计算误差是否会发生变化还有待研究。本文在高阶精度差分格式下通过浅水波方程对跳点网格和非跳点网格的频散关系进行理论分析和数值试验，主要得到以下结论：（1）在低波数区跳点网格的频散关系基本不随计算精度的提高而变化，但是非跳点网格下的频散关系则随着计算精度的提高而更加接近真实解。在四阶精度下，非跳点网格的频散关系已经非常接近跳点网格。（2）差分精度提高以后，在高波数区非跳点网格仍然存在频率极大值，而且极值中心随着计算精度的提高而逐渐向更高波数区移动。跳点网格在计算精度提高以后高波数区的频率仍然随波数单调增加，且更接近真实解。（3）在高阶精度非跳点网格模拟试验的基础上，结合高阶扩散项对高频短波进行滤除，可以得到与二阶精度跳点网格相接近的模拟结果。总之，在高阶精度有限差分方案下利用非跳点网格构造模式动力框架是一种比较可行的做法。Abstract: Although the nonstaggered grid has better physical consistency than the staggered grid, it is still not widely used, mainly because of its poor accuracy for simulating the geostrophic adjustment process under the second-order accuracy difference scheme. However, for higher-order finite-difference schemes, the validity of the above conclusion is still unknown. In this paper, we present a theoretical analysis and a numerical test for shallow water equation under high-order difference schemes. The analysis and test results are as follows. (1) For a low wavenumber, the dispersion of a staggered grid does not change with the accuracy of the difference scheme, while the dispersion of nonstaggered grid is significantly improved, and the dispersion of both grids become very close under the fourth-order scheme. (2) The maximum frequency of nonstaggered grid still exists under high-order difference schemes, and it moves toward a higher wavenumber as the accuracy of the difference scheme increases. The frequency of the staggered grid monotonically increases with the wavenumber and gets closer to the real solution under high-order difference schemes. (3) When the high-frequency noise is removed by explicitly adding a diffusion term, the pros and cops to grid staggering choices diminish with high-order schemes. In general, the nonstaggered grid is an attractive choice for the discretization of the dynamical frame in numerical models under high-order difference scheme.
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Key words:
- Nonstaggered grid /
- High order accuracy /
- Finite difference /
- Shallow water equation
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图 1 二维浅水波方程中无量纲频率（等值线）和波数的真实关系，图中kd/pi和ld/pi分别表示x方向和y方向的无量纲波数（pi为圆周率）
Figure 1. Contours of the nondimensional frequency (contour) as a function of the nondimensional horizontal wavenumbers for a true solution of a shallow water equation.
$ kd/pi $ and$ ld/pi $ are the nondimensional wavenumbers at$ x $ and$ y $ directions, respectively, pi is the ratio of the circumference of a circle to its diameter.
图 2 浅水波试验中的二维网格变量分布：（a）A网格；（b）C网格
Figure 2. Distributions of variables in two-dimensional grid for the test of a shallow water equation: (a) A grid; (b) C grid
图 3 不同网格和差分精度下的无量纲频率（等值线）：（a）C网格＋二阶精度；（b）C网格＋四阶精度；（c）C网格＋六阶精度；（d）A网格＋二阶精度；（e）A网格＋四阶精度；（f）A网格＋六阶精度。深蓝色方框表示波长大于4倍格距的波数范围
Figure 3. Nondimensional frequency (Contours) under different grids and accuracy of differential computation : (a) C grid ＋ 2nd order; (b) C grid ＋ 4th order; (c) C grid ＋ 6th order; (d) A grid ＋ 2nd order; (e) A grid ＋ 4th order; (f) A grid ＋ 6th order. The blue square represents wavelength exceeding four-grid interval
图 4 当t=300 s时在跳点网格和非跳点网格下的波长为十倍格距的重力波模拟：（a）test-C-2nd-10；（b）test-A-2nd-10；（c）test-A-4th-10；（d）test-A-6th-10
Figure 4. Simulation of gravity wave whose wavelength is set as 10 grid spacing using staggered and nonstaggered grids at t = 300 s: (a) test-C-2nd-10; (b) test-A-2nd-10; (c) test-A-4th-10; (d) test-A-6th-10
图 5 （a）当t=300 s时test-A-6th-10-f试验对重力波的模拟；（b）不同方案下波长为十倍格距的重力波均方根误差随时间的增长
Figure 5. (a) Simulation of gravity wave for test-A-6th-10-f at t = 300 s; (b) the growth of the root-mean-squared error underdifferent schemes for the gravity wave whose wavelength is set as 10 grid spacing.
图 6 当t=300 s时在跳点网格和非跳点网格下的波长为四倍格距的重力波模拟：（a）test-C-2nd-4；（b）test-A-2nd-4；（c）test-A-4th-4；（d）test-A-6th-4
Figure 6. Simulation of the gravity wave whose wavelength is set as 4 grid spacing using staggered grid and unstaggered at t=300s: (a) test-C-2nd-4; (b) test-A-2nd-4; (c) test-A-4th-4; (d) test-A-6th-4
图 7 当t=300 s时在跳点网格和非跳点网格下的四倍格距波模拟（在滤除高频噪音的情况下）：（a）test-C-2nd-4-f；（b）test-A-6th-4-f
Figure 7. Simulation of gravity wave whose wavelength is set as 4 grid spacing with staggered and nonstaggered grids at t = 300 s (with the high frequency noise being filtered out): (a) test-C-2nd-4-f; (b) test-A-6th-4-f
图 8 跳点网格下不同精度算法对重力波模拟：（a）test-C-4th-10；（b）test-C-6th-10；（c）test-C-4th-4；（d）test-C-6th-4
Figure 8. Simulation of gravity wave with different accuracies under staggered grids: (a) test-C-4th-10; (b) test-C-6th-10; (c) test-C-4th-4; (d) test-C-6th-4
表 1 跳点和非跳点网格下的不同精度差分格式
Table 1. Difference schemes with different accuracies under staggered and nonstaggered grids
网格 差分精度 差分格式 非跳点网格 二阶精度 ${\left(\displaystyle\frac{\partial f}{\partial x}\right)}_{i}=\displaystyle\frac{ {f}_{i+1}-{f}_{i-1} }{2\Delta x}$ 四阶精度 ${\left(\displaystyle\frac{\partial f}{\partial x}\right)}_{i}=\displaystyle\frac{ {f}_{i-2}-8{f}_{i-1}+8{f}_{i+1}-{f}_{i+2} }{12\Delta x}$ 六阶精度 ${\left(\displaystyle\frac{\partial f}{\partial x}\right)}_{i}=\displaystyle\frac{-{f}_{i-3}+9{f}_{i-2}-45{f}_{i-1}+45{f}_{i+1}-9{f}_{i+2}+{f}_{i+3} }{60\Delta x}$ 跳点网格 二阶精度 ${\left(\displaystyle\frac{\partial f}{\partial x}\right)}_{i}=\displaystyle\frac{ {f}_{i+0.5}-{f}_{i-0.5} }{\Delta x}$ 四阶精度 ${\left(\displaystyle\frac{\partial f}{\partial x}\right)}_{i}=\displaystyle\frac{ {f}_{i-1.5}-27{f}_{i-0.5}+27{f}_{i+0.5}-{f}_{i+1.5} }{24\Delta x}$ 六阶精度 ${\left(\displaystyle\frac{\partial f}{\partial x}\right)}_{i}=\displaystyle\frac{-9{f}_{i-2.5}+125{f}_{i-1.5}-2250{f}_{i-0.5}+2250{f}_{i+0.5}-125{f}_{i+1.5}+9{f}_{i+2.5} }{1920\Delta x}$ 
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表 2 A网格和C网格下不同精度差分格式的频率方程
Table 2. Frequencies equation of Finite-difference grids A and C with different accuracies
网格 差分精度 频率方程 A网格 二阶精度 ${\left(\displaystyle\frac{\omega }{ {f}_{0} }\right)}^{2}=$1$+{\left(\displaystyle\frac{ {L}_{0} }{d}\right)}^{2}\left[{\left(\sin\left(ld\right)\right)}^{2}+{\left(\sin\left(kd\right)\right)}^{2}\right]$ 四阶精度 ${\left(\displaystyle\frac{\omega }{ {f}_{0} }\right)}^{2}=$1$+{\left(\displaystyle\frac{ {L}_{0} }{d}\right)}^{2}\left[{\left(\displaystyle\frac{-\sin\left(2ld\right)+8\sin\left(ld\right)}{6}\right)}^{2}+{\left(\displaystyle\frac{-\sin\left(2kd\right)+8\sin\left(kd\right)}{6}\right)}^{2}\right]$ 六阶精度 ${\left(\displaystyle\frac{\omega }{ {f}_{0} }\right)}^{2}=$1$+{\left(\displaystyle\frac{ {L}_{0} }{d}\right)}^{2}\left[{\left(\displaystyle\frac{\sin\left(3ld\right)-9\sin\left(2ld\right)+45\sin\left(ld\right))}{30}\right)}^{2}+{\left(\displaystyle\frac{\sin\left(3kd\right)-9\sin\left(2kd\right)+45\sin\left(kd\right)}{30}\right)}^{2}\right]$ C网格 二阶精度 ${\left(\displaystyle\frac{\omega }{ {f}_{0} }\right)}^{2}={(\cos(0.5kd)\cos(0.5ld\left)\right)}^{2}+{4\left(\displaystyle\frac{ {L}_{0} }{d}\right)}^{2}\left[{\left(\sin\left(0.5ld\right)\right)}^{2}+{\left(\sin\left(0.5kd\right)\right)}^{2}\right]$ 四阶精度 ${\left(\displaystyle\frac{\omega }{ {f}_{0} }\right)}^{2}={(\cos(0.5kd)\cos(0.5ld\left)\right)}^{2}+{\left(\frac{ {L}_{0} }{d}\right)}^{2}\left[{\left(\displaystyle\frac{-\sin\left(1.5ld\right)+27\sin\left(0.5ld\right)}{12}\right)}^{2}+{\left(\displaystyle\frac{-\sin\left(1.5kd\right)+27\sin\left(0.5kd\right)}{12}\right)}^{2}\right]$ 六阶精度 $\begin{array}{l}{\left(\displaystyle\frac{\omega }{ {f}_{0} }\right)}^{2}={(\cos(0.5kd)\cos(0.5ld\left)\right)}^{2}+{\left(\displaystyle\frac{ {L}_{0} }{d}\right)}^{2}\\\left[{\left(\displaystyle\frac{9\sin\left(2.5ld\right)-125\sin\left(1.5ld\right)+2250\sin\left(0.5ld\right)}{960d}\right)}^{2}+{\left(\displaystyle\frac{9\sin\left(2.5kd\right)-125\sin\left(1.5kd\right)+2250\sin\left(0.5kd\right)}{960d}\right)}^{2}\right]\end{array}$
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表 3 试验设计
Table 3. Design of experiment
试验名称 网格 差分精度 波长 是否滤波 Test-A-2nd-10 A网格 二阶 10倍格距 否 Test-A-4th-10 四阶 否 Test-A-6th-10 六阶 否 Test-A-6th-10-f 六阶 是 Test-A-2nd-4 二阶 4倍格距 否 Test-A-4th-4 四阶 否 Test-A-6th-4 六阶 否 Test-A-6th-4-f 六阶 是 Test-C-2nd-10 C网格 二阶 10倍格距 否 Test-C-4th-10 四阶 否 Test-C-6th-10 六阶 否 Test-C-2nd-4 二阶 4倍格距 否 Test-C-2nd-4-f 二阶 是 Test-C-4th-4 四阶 否 Test-C-6th-4 六阶 否
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