Abstract
We introduce an efficient split finite element (FE) discretization of a
y-independent (slice) model of the rotating shallow water equations. The study
of this slice model provides insight towards developing schemes for the full 2D
case. Using the split Hamiltonian FE framework (Bauer, Behrens and Cotter,
2019), we result in structure-preserving discretizations that are split into
topological prognostic and metric-dependent closure equations. This splitting
also accounts for the schemes' properties: the Poisson bracket is responsible
for conserving energy (Hamiltonian) as well as mass, potential vorticity and
enstrophy (Casimirs), independently from the realizations of the metric closure
equations. The latter, in turn, determine accuracy, stability, convergence and
discrete dispersion properties. We exploit this splitting to introduce
structure-preserving approximations of the mass matrices in the metric
equations avoiding to solve linear systems. We obtain a fully
structure-preserving scheme with increased efficiency by a factor of two.