Numerical wave propagation for the triangular P1DG-P2 finite element pair

The f -plane and β-plane wave propagation properties are examined for discretisations of the linearized rotating shallow-water equations using the P1DG-P2 finite element pair on arbitrary triangulations in planar geometry. A discrete Helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations. In the f -plane (planar geometry, Coriolis force independent of space) case, this decomposition is used to obtain decoupled equations for the geostrophic modes, the inertia-gravity modes, and the inertial oscillations. As has been noticed previously, the geostrophic modes are steady. The Helmholtz decomposition is used to show that the resulting inertia-gravity wave equation is third-order accurate in space. In general the P1DG-P2 finite element pair is second-order accurate, so this leads to very accurate wave propagation. It is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency f , and which do not propagate. A restriction of the P1DG velocity space is proposed in which these modes are not present, leading to a finite element discretisation which is completely free of spurious modes. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised P1DG-P2 equations in the β-plane (planar geometry, Coriolis force linear in space) case resulting in a Rossby wave equation which is also third-order accurate. This means that the dispersion relation for the wave propagation is very accurate; an illustration of this is provided by a numerical dispersion analysis in the case of a triangulation consisting of equilateral triangles.