Analysis of a mixed finite‐element pair proposed for an atmospheric dynamical core
Analysis of a mixed finite‐element pair proposed for an atmospheric dynamical core
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DOI：
10.1002/qj.2028
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发表时间：
2013-07
影响因子：
8.9
通讯作者：
A. Staniforth;T. Melvin;C. Cotter
中科院分区：
文献类型：
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作者：
A. Staniforth;T. Melvin;C. Cotter
我们提出了用于在一个维度上应用于线性浅水方程的P2 -P1DG有限元对的数值分析。最近提出了用于具有准均匀网格的大气动力学的水平离散已显示有限元对具有许多理想的特性，可将C网格的特性扩展到非正交的四边形和三角网格，包括F平面上的固定地遗传模式，以及速度与自由度的比率为2：1 （对于没有虚假模式分支的必要条件），在没有科里奥利的情况下，适当的物理数值波传播也很重要。 - p1dg。波长接近两个元素，这是一个潜在的问题，因为它增加了需要过滤的波数空间。通过对方程式的小修改（即部分块状矩阵）的删除，以使方案的其他有利属性不受影响。保存能量保存（修改了离散动能的定义）。二维方案。
We present a numerical dispersion analysis for the P2 − P1DG finite‐element pair applied to the linear shallow‐water equations in one dimension. The aim is to provide insight into the numerical dispersion properties of the RT1 and BDFM1 finite‐element pairs in two dimensions, which have recently been proposed for horizontal discretisations of atmospheric dynamical cores with quasi‐uniform grids. This is achieved via analysis of a one‐dimensional RT1 element. Whilst these finite‐element pairs have been shown to have many desirable properties that extend properties of the C grid to non‐orthogonal quadrilateral and triangular grids, including stationary geostrophic modes on the f plane, and a 2:1 ratio of velocity to pressure degrees of freedom (a necessary condition for the absence of spurious mode branches), it is also important to have appropriately physical numerical wave propagation. In the absence of Coriolis force, we compute the group velocity for P2 − P1DG. We find that, as well as dropping to zero at the grid‐scale, which also occurs for the C‐grid finite‐difference method, the group velocity also drops to zero in a narrow band around kh = π which corresponds to eigenmodes with a wavelength close to two element widths. This is a potential problem because it increases the amount of wavenumber space that needs to be filtered. In this one‐dimensional case, we find that this particular issue can be removed by a small modification of the equations, namely partially lumping the mass matrix, in such a way that the other favourable properties of the scheme are not affected. We discuss various symmetric and asymmetric modifications of the mass matrix, and show that both such modifications preserve energy conservation (having modified the definition of discrete kinetic energy). Finally we illustrate our findings with numerical experiments, and discuss the potential to extend this modification to two‐dimensional schemes.