Analysis of Numerically Induced Oscillations in 2D Finite‐Element Shallow‐Water Models Part I: Inertia‐Gravity Waves

Abstract

The numerical approximation of shallow-water models is a delicate problem. For most of the discretization schemes, the coupling between the momentum and the continuity equations usually leads to anomalous dispersion in the representation of fast waves. A dispersion relation analysis is employed here to ascertain the presence and determine the form of spurious modes as well as the dispersive nature of the finite-element Galerkin mixed formulation of the two-dimensional linearized shallow-water equations. Nine popular finite-element pairs are considered using a variety of mixed interpolation schemes. For each pair the frequency or dispersion relation is obtained and analyzed, and the dispersion properties are compared analytically and graphically with the continuous case. It is shown that certain choices of mixed interpolation schemes may lead to significant phase and group velocity errors and spurious solutions in the calculation of fast waves. The $P^{NC}_{1} - P^{}_{1}$ and RT0 pairs are identified as a promising compromise, provided the grid resolution is high relative to the Rossby radius of deformation for the RT0 element. The numerical solutions of two test problems to simulate fast waves are in good agreement with the analytical results.