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CN 11-1768/O4

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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
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中国气象局广州热带海洋气象研究所

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Nonstaggered grid test under high order finite difference scheme: based on 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 staggered grid, it is still not widely used mainly because its poor accuracy for simulating the geostrophic adjustment process under 2nd order accuracy difference scheme. However, weather this conclusion remains valid under higher order finite difference scheme is still unknown. In this paper we conduct theoretical analysis and numerical test for shallow water equation under high order difference schemes, it is found that: (1) for low wave number, the dispersion of staggered grid does not change with the accuracy of difference scheme, while the dispersion of unstaggered grid is improved obviously, and the dispersion of both grid get very close under 4th order scheme. (2) the maximum frequency of unstaggered grid still exist under high order difference scheme, and it move towards higher wave number as the accuracy of difference scheme get higher. The frequency of staggered grid is monotonically increasing with wavenumber and gets more close to real solution under high order difference scheme. (3) when the high frequency noise was removed with explicitly adding a diffusion term, the pros and cops to grid staggering choices diminish with high-order schemes. In general, the unstaggered grid is an attractive choose for the discretization of the dynamical frame in numerical model under high-order difference scheme.

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历史
  • 收稿日期:2019-10-23
  • 最后修改日期:2019-12-09
  • 录用日期:2020-09-02
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