paper

Numerical wave propagation for the triangular - finite element pair

arXiv:1001.2925 · doi:10.1016/j.jcp.2010.12.024

Abstract

Inertia-gravity mode and Rossby mode dispersion properties are examined for discretisations of the linearized rotating shallow-water equations using the - 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 -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 , and which do not propagate. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised - equations in the -plane case, resulting in a Rossby wave equation which is also third-order accurate.

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