## 1. Introduction

In the context of the expected evolution of massively parallel supercomputer architectures, Staniforth and Thuburn (2012) have recently reviewed and assessed horizontal grids for modelling the atmosphere over the sphere. They concluded that all grids proposed to date have known problems or issues that merit further investigation. Traditional latitude–longitude-based grids lead to data communication bottlenecks due to resolution clustering around the two poles. However, although discretisations on alternative, quasi-uniform, grids reduce the severity of this problem, they are prone to other drawbacks such as computational modes and grid imprinting. Computational modes are potentially very serious, since they may be excited in realistic applications by boundary conditions, nonlinearity, physical forcing, and data assimilation.

Much of the current thrust of research on atmospheric dynamical cores is focused on discretisations that employ either a subdivided icosahedral grid, or a cubed-sphere grid (Lauritzen *et al.*, 2011). In a recent development, Cotter and Shipton (2012) have proposed the use of some particular families of mixed finite elements as a means of improving the horizontal discretisation of atmospheric dynamical cores on such quasi-uniform grids. They demonstrate that these mixed elements preserve the following desirable properties of the C-grid method when applied to linear barotropic wave propagation:

- 1.energy conservation;
- 2.mass conservation;
- 3.absence of spurious pressure modes;
- 4.steady geostrophic modes on an
*f*-plane; and - 5.the flexibility to adjust the balance of velocity degrees of freedom to pressure degrees of freedom, so as to achieve a 2:1 ratio.

Regarding the last property, it was shown by Cotter and Shipton (2012) that the 2:1 ratio is a necessary condition for absence of spurious mode branches for this family of mixed finite-element methods.

Two examples of mixed finite elements are specifically recommended: the recently developed modified Raviart–Thomas (RT) element on quadrilaterals, and the Brezzi–Douglas–Fortin–Marini (BDFM) element on triangles. The lowest-degree RT finite-element space is the mixed finite-element analogue of the C grid: the pressure space has piecewise constant (discontinuous) functions, and the velocity fields are bilinear, but constrained to have constant, continuous normal components on element edges. This mixed element shows promise for discretisation on a cubed sphere.

The second of Cotter and Shipton (2012)'s proposed mixed elements is based on triangles, and offers promise for discretisations on triangularly subdivided icosahedral grids. The lowest possible order for triangular elements that allows a 2:1 ratio is the BDFM1 element pair (piecewise linear and discontinuous for pressure, piecewise quadratic with continuous normal components for velocity, with the constraint that normal components are linear on edges). In contrast to the quadrilateral case, where the lowest possible order is RT0, which is built from constant and linear functions, the velocity representation for BDFM1 contains quadratic functions (although not all of them). Therefore a key question is whether or not the inclusion of quadratic functions for these triangular elements is likely to adversely affect numerical dispersion: a good representation of group velocity is of crucial importance since it controls energy propagation. In this article, we take steps towards understanding the dispersion properties of BDFM1 by studying the RT1 space on quadrilaterals, which is the quadrilateral analogue of BDFM1 and also has quadratic functions in the velocity expansion. If the RT1 space on quadrilaterals turns out to suffer from poor dispersion properties, then it seems likely that BDFM1 will also inherit the same poor dispersion properties. An important advantage of this approach is that, for the rectangular case, the problem can, under the assumption of variation in only one direction, be reduced to a one-dimensional (1D) problem, thereby greatly facilititating analysis tractability.

A mixture of discontinuous and continuous finite-element spaces has long been employed for fluid dynamical problems, as one way of satisfying the Babuska–Brezzi stability condition. The *P*2 − *P*1_{DG} scheme considered here in one dimension can be extended to several different schemes in two dimensions depending on the continuity and extra degrees of freedom in the extra direction. The most obvious extension is to take *Q*2 − *Q*1_{DG} on quadrilaterals in two dimensions, which forms part of the *Q*(*k*) − *Q*(*k* − 1)_{DG} high-order family that was proposed for ocean modelling by Iskandarani *et al.* (1995). In that article, a spectral element approach was used in which incomplete (but consistent) quadrature leads to a diagonal mass matrix. In the context of engineering flows, the related *P*(*k*) − *P*(*k* − 2)_{DG} space on triangles was introduced for the Navier–Stokes equations by Deville *et al.* (2002). A fully discontinuous *Qk*_{DG} − *Qk*_{DG} approach was introduced by Giraldo *et al.* (2002), and extended to *Pk*_{DG} − *Pk*_{DG} triangles by Giraldo (2006). The quest to find schemes without spurious modes has led to the consideration of partially continuous spaces for velocity, including the Brezzi–Douglas–Marini (BDM), the BDFM, and RT spaces, which are all described by Brezzi and Fortin (1991). The dual scheme *P*1_{DG} − *P*2 was proposed by Cotter *et al.* (2009a). The dispersion properties of a number of these schemes have been analysed, including equal-order DG schemes (Ainsworth *et al.*, 2006), RT and BDM schemes (Rostand and Le Roux, 2008), and *P*1_{DG} − *P*2 by Cotter and Ham (2011) and Le Roux (2012). Dispersion relations for a number of other finite-element spaces were analysed by Le Roux *et al.* (2007) and Le Roux *et al.* (2008). A number of spaces were analysed by inspecting the matrix structure in Le Roux *et al.* (2005), and five discontinuous and partially discontinuous mixed finite-element spaces were investigated numerically by Comblen *et al.* (2010). The approach of modifying the mass matrix to improve dispersion properties was investigated by Gassner and Kopriva (2011), although the types of lumping were quadrature-based unlike the partial lumping discussed in this article.

The continuous 1D linear shallow-water equations used in this study are given in section 2. The mixed-element discretisations of these equations are derived in appendices and summarised in section 3. A dispersion analysis of a subset of these equations, the gravity wave equations, is presented in section 4. This reveals a dispersion problem for waves with wavelengths equal or close to twice the element width. A means of addressing this issue for the considered 1D problem is described in section 5. Illustrative numerical integrations of the 1D gravity wave equations are presented in section 6. Energy conservation for the proposed solution is then examined in section 7. Finally, conclusions are drawn in section 8, including a brief discussion of outstanding issues.