Analysis of a mixed finite-element pair proposed for an atmospheric dynamical core

We present a numerical dispersion analysis for the P2 - P1(DG) 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 - P1(DG). 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 = 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.