Abstract
Construction of high-order difference schemes based on Taylor series expansion has long been a hot topic in computational mathematics, while its application in comprehensive weather models is still very rare. Here, the properties of high-order finite difference schemes are studied based on idealized numerical testing, for the purpose of their application in the Global/Regional Assimilation and Prediction System (GRAPES) model. It is found that the pros and cons due to grid staggering choices diminish with higher-order schemes based on linearized analysis of the one-dimensional gravity wave equation. The improvement of higher-order difference schemes is still obvious for the mesh with smooth varied grid distance. The results of discontinuous square wave testing also exhibits the superiority of high-order schemes. For a model grid with severe non-uniformity and non-orthogonality, the advantage of high-order difference schemes is inapparent, as shown by the results of two-dimensional idealized advection tests under a terrain-following coordinate. In addition, the increase in computational expense caused by high-order schemes can be avoided by the precondition technique used in the GRAPES model. In general, a high-order finite difference scheme is a preferable choice for the tropical regional GRAPES model with a quasi-uniform and quasi-orthogonal grid mesh.
摘要
基于泰勒展开构造高阶精度有限差分方案很早就成为了计算数学界的热门话题, 但是它在成熟的数值天气预报模式中仍然很少被应用. 为了将高阶精度有限差分方案应用于 GRAPES (Global/Regional Assimilation and Prediction System) 模式, 本文通过理想试验对高阶精度有限差分方案的相关性质进行了分析. 一维重力波的线性分析结果表明, 在高阶精度差分方案下不同跳点网格方案的影响很小. 一维平流试验结果表明: 对于网格距离缓慢变化的非均匀网格, 高阶差分方案的改进效果仍然十分明显, 不连续方波试验的结果也表明了高阶方案的优越性. 在地形追随坐标系下的二维质量平流试验结果表明, 当模式网格的非均匀性和非正交性变得很明显时, 高阶方案的改进效果变得不是很明显. 通过在GRAPES模式中采用预条件技术, 可以有效地避免高阶方案引起计算量增加的问题. 总而言之, 对于准均匀和准正交网格的热带区域GRAPES模式来说, 发展基于高阶精度有限差分方案的动力框架是一种可行的选择.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. U1811464). We thank the two anonymous reviewers and the editors for their comments, which greatly improved this paper.
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Article Highlights
• It is found that the pros and cons due to grid staggering choices diminish with higher-order difference schemes.
• The improvement of higher-order difference schemes is still obvious in non-uniform grid and discontinuous square wave tests.
• For a model grid with severe non-uniformity and non-orthogonality, the advantage of high-order difference schemes will be inapparent.
• The increase in computational expense caused by high-order schemes can be avoided by the precondition technique in the GRAPES model.
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Xu, D., Chen, D. & Wu, K. Properties of High-Order Finite Difference Schemes and Idealized Numerical Testing. Adv. Atmos. Sci. 38, 615–626 (2021). https://doi.org/10.1007/s00376-020-0130-7
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DOI: https://doi.org/10.1007/s00376-020-0130-7