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Inertia-gravity mode and Rossby mode dispersion properties are examined fordiscretisations of the linearized rotating shallow-water equations using the$P1_{DG}$-$P2$ finite element pair on arbitrary triangulations in planargeometry. A discrete Helmholtz decomposition of the functions in the velocityspace based on potentials taken from the pressure space is used to provide acomplete description of the numerical wave propagation for the discretisedequations. In the $f$-plane case, this decomposition is used to obtaindecoupled equations for the geostrophic modes, the inertia-gravity modes, andthe inertial oscillations. As has been noticed previously, the geostrophicmodes are steady. The Helmholtz decomposition is used to show that theresulting inertia-gravity wave equation is third-order accurate in space. Ingeneral the \pdgp finite element pair is second-order accurate, so this leadsto very accurate wave propagation. It is further shown that the only spuriousmodes supported by this discretisation are spurious inertial oscillations whichhave frequency $f$, and which do not propagate. The Helmholtz decompositionalso allows a simple derivation of the quasi-geostrophic limit of thediscretised $P1_{DG}$-$P2$ equations in the $\beta$-plane case, resulting in aRossby wave equation which is also third-order accurate.

Numerical wave propagation for the triangular $P1_{DG}$-$P2$ finite element pair