Dispersion analysis of the spectral element method
Dispersion analysis of the spectral element method
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DOI：
10.1002/qj.1906
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发表时间：
2012-10
影响因子：
8.9
通讯作者：
T. Melvin;A. Staniforth;J. Thuburn
中科院分区：
文献类型：
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作者：
T. Melvin;A. Staniforth;J. Thuburn
谱元法 (SEM) 具有（具有精确的时间积分）局部和全局质量、能量和位涡守恒的理想属性。它在大规模并行计算机上也能很好地扩展。大气动力核心数值方法的另一个理想属性是它应该具有良好的数值色散特性，以便准确地表示波传播和调整过程。分析了 SEM 在单向波动方程中的应用，以深入了解其作为谱阶函数的色散特性。对于最低阶光谱截断（线性），SEM 离散化在形式上等效于 Arakawa A 网格上的中心二阶有限差分。因此，它具有较差的色散特性，包括光谱的短波长一半的能量传播方向错误。将 SEM 的光谱截断增加到二次可以改善其光谱长波长部分的色散特性，但短波长部分的能量传播方向错误的问题仍然存在。进一步增加谱截断的阶数不仅不能解决小尺度下能量传播不良的问题，而且还引入了新的问题，包括可以表示的频率谱中的间隙，以及单元边界附近的本征模结构的局域化。数值积分证实这些 SEM 色散特性会导致相反的群速度并导致光谱元素边界处的网格印记。版权所有 © 2012 英国皇家气象学会和英国皇家版权所有，英国气象局
The spectral element method (SEM) has (with exact time integration) the desirable attribute of locally and globally conserving mass, energy and potential vorticity. It also scales well on massively parallel computers. Another desirable attribute of a numerical method for an atmospheric dynamical core is that it should have good numerical dispersion properties in order to accurately represent wave propagation and adjustment processes. Application of the SEM to the one‐way wave equation is analysed to provide insight into its dispersion properties as a function of spectral order. For the lowest‐order spectral truncation (linear) the SEM discretisation is formally equivalent to centred second‐order finite differences on an Arakawa A grid. It consequently shares its poor dispersion properties, including energy propagation in the wrong direction for the short‐wavelength half of the spectrum. Increasing the spectral truncation of the SEM to quadratic improves its dispersion properties for the long‐wavelength part of the spectrum, but the problem of energy propagation in the wrong direction for the short‐wavelength part remains. Further increasing the order of the spectral truncation not only fails to address the poor energy propagation at small scales, but also introduces new problems, including gaps in the spectrum of frequencies that can be represented, and localisation of eigenmode structures near element boundaries. Numerical integrations confirm that these SEM dispersion properties lead to reversed group velocities and to grid imprinting at spectral element boundaries. Copyright © 2012 Royal Meteorological Society and British Crown Copyright, the Met Office