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

TL;DR

This paper analyzes the wave propagation properties of a specific finite element discretization of the rotating shallow-water equations, demonstrating high accuracy and identifying spurious inertial oscillations.

Contribution

It introduces a Helmholtz decomposition-based analysis for the $P1_{DG}$-$P2$ finite element pair, revealing its accuracy and mode properties in wave simulations.

Findings

Inertia-gravity wave equation is third-order accurate in space.

Spurious modes are limited to non-propagating inertial oscillations.

Discretization accurately captures geostrophic and Rossby wave dynamics.

Abstract

Inertia-gravity mode and Rossby mode dispersion properties are examined for discretisations of the linearized rotating shallow-water equations using the $P1_{DG}$-$P2$ 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 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 \pdgp 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. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised $P1_{DG}$-$P2$ equations in the $\beta$-plane case, resulting in a Rossby wave equation which is also third-order accurate.