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高阶精度有限差分方案下的非跳点网格试验:基于浅水波方程

徐道生 陈德辉 吴凯昕

徐道生, 陈德辉, 吴凯昕. 2021. 高阶精度有限差分方案下的非跳点网格试验:基于浅水波方程[J]. 大气科学, 45(3): 1−11 doi: 10.3878/j.issn.1006-9895.2007.19236
引用本文: 徐道生, 陈德辉, 吴凯昕. 2021. 高阶精度有限差分方案下的非跳点网格试验:基于浅水波方程[J]. 大气科学, 45(3): 1−11 doi: 10.3878/j.issn.1006-9895.2007.19236
XU Daosheng, CHEN Dehui, WU Kaixin. 2021. Nonstaggered Grid under High-Order Finite-Difference Scheme—Analysis Based on a Shallow Water Equation [J]. Chinese Journal of Atmospheric Sciences (in Chinese), 45(3): 1−11 doi: 10.3878/j.issn.1006-9895.2007.19236
Citation: XU Daosheng, CHEN Dehui, WU Kaixin. 2021. Nonstaggered Grid under High-Order Finite-Difference Scheme—Analysis Based on a Shallow Water Equation [J]. Chinese Journal of Atmospheric Sciences (in Chinese), 45(3): 1−11 doi: 10.3878/j.issn.1006-9895.2007.19236

高阶精度有限差分方案下的非跳点网格试验:基于浅水波方程

doi: 10.3878/j.issn.1006-9895.2007.19236
基金项目: 国家重点研发专项项目2018YFC1506901,国家自然科学基金项目41705035,广东省科技计划项目2017A020219005
详细信息
    作者简介:

    徐道生,男,1985年出生,博士,主要从事数值预报研究。E-mail: dsxu@gd121.cn

  • 中图分类号: P456.7

Nonstaggered Grid under High-Order Finite-Difference Scheme—Analysis Based on a Shallow Water Equation

Funds: Special Foundation for State Major Basic Research Program of China (Grant 2018YFC1506900), National Natural Science Foundation of China (Grant 41705035), Science and Technology Planning Project of Guangdong Province (Grant 2017A020219005)
  • 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。
  • 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/pild/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  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  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  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${\left(\displaystyle\frac{\partial f}{\partial x}\right)}_{i}=\displaystyle\frac{-9{f}_{i-2.5}+125{f}_{i-1.5}-2250{f}_{i-0.5}+2250{f}_{i+0.5}-125{f}_{i+1.5}+9{f}_{i+2.5} }{1920\Delta x}$
    下载: 导出CSV

    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$\begin{array}{l}{\left(\displaystyle\frac{\omega }{ {f}_{0} }\right)}^{2}={(\cos(0.5kd)\cos(0.5ld\left)\right)}^{2}+{\left(\displaystyle\frac{ {L}_{0} }{d}\right)}^{2}\\\left[{\left(\displaystyle\frac{9\sin\left(2.5ld\right)-125\sin\left(1.5ld\right)+2250\sin\left(0.5ld\right)}{960d}\right)}^{2}+{\left(\displaystyle\frac{9\sin\left(2.5kd\right)-125\sin\left(1.5kd\right)+2250\sin\left(0.5kd\right)}{960d}\right)}^{2}\right]\end{array}$
    下载: 导出CSV

    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
    Test-A-4th-10 四阶
    Test-A-6th-10 六阶
    Test-A-6th-10-f 六阶
    Test-A-2nd-4 二阶 4倍格距
    Test-A-4th-4 四阶
    Test-A-6th-4 六阶
    Test-A-6th-4-f 六阶
    Test-C-2nd-10 C网格 二阶 10倍格距
    Test-C-4th-10 四阶
    Test-C-6th-10 六阶
    Test-C-2nd-4 二阶 4倍格距
    Test-C-2nd-4-f 二阶
    Test-C-4th-4 四阶
    Test-C-6th-4 六阶
    下载: 导出CSV
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  • 收稿日期:  2019-10-23
  • 录用日期:  2020-09-02
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