Splitting horizontal and vertical polynomial order in a compatible finite element discretisation for numerical weather prediction

Abstract

The accurate and efficient representation of atmospheric dynamics remains a central challenge in numerical weather prediction. A particular difficulty arises from the strong anisotropy of the atmosphere, in which horizontal and vertical motions occur on very different length scales, motivating numerical discretisations that can reflect this structure. In this study, we introduce a compatible finite element discretisation of the compressible Boussinesq and compressible Euler equations in which the horizontal and vertical polynomial orders are treated independently. The split-order discretisation is constructed using a tensor-product framework that preserves the discrete de Rham complex and associated mimetic properties. Its wave-propagation characteristics are examined through a discrete dispersion analysis that extends previous analyses to configurations with differing horizontal and vertical polynomial orders. The results show that increasing horizontal order improves the representation of gravity waves at low and intermediate wavenumbers, while increasing vertical order can degrade dispersion accuracy near the grid scale and introduce spectral gaps. A series of idealised numerical experiments, including gravity-wave propagation, advective transport, mountain-wave flow, and a global baroclinic-wave test, is used to assess the scheme's accuracy and convergence properties. These experiments demonstrate that increasing the polynomial order in the dominant direction of motion improves convergence, and that increasing the horizontal order yields the greatest gain in accuracy under typical atmospheric conditions. The results indicate that split-order compatible finite element discretisations provide a viable alternative for controlling accuracy and numerical behaviour in atmospheric dynamical cores.

Figures (11)

• Figure 1: Diagrams of the one-dimensional finite element spaces used to build the two-dimensional spaces. Here, DG1 refers to discontinuous order 1 (linear) polynomials over the cell, CG2 continuous order 2 (quadratic) polynomials, etc. Dots represent degrees of freedom (DoF).
• Figure 2: Analytical dispersion relation for the acoustic (left) and gravity (right) modes of the linear compressible Boussinesq equations.
• Figure 3: Locations of degrees of freedom for the $(1,0)$ case for the horizontal velocity field $(u)$ shown in the top left, followed by the vertical velocity field $(w)$ in the top right, the pressure field $(p)$ in the bottom left, and the buoyancy field $(b)$ in the bottom right. The node numbering can be related to node locations using equations \ref{['eqn:number_guide']}. Nodes on a cell edge indicate continuity between cells, and an arrow across an edge indicates normal continuity.
• Figure 4: Discrete dispersion relations for the $(1, 1)$, $(1, 0)$ and $(0, 1)$ finite element cases, for the scaled wavenumbers $\tilde{k}$, $\tilde{l}$$\in[0, \frac{2\pi}{\Delta x}] \times [0, \frac{2\pi}{\Delta z}]$, where we arbitarity take $(\Delta x, \Delta z) = (1000, 1000)\unit{\meter}$. To the right of each mode is a plot showing the percentage error between the discrete and analytical dispersion relations.
• Figure 5: Buoyancy perturbation of the Skamarock and Klemp gravity wave for the compressible boussinesq equations, test case at time t=3600s. Contours are between $[-3, 3]\times e-3m\per\square s$ with a spacing of $0.5\times e-3m\per\square s$ and are the same for each case.