Modal analysis on unstructured meshes of the dispersion properties of the P1NC-P1 pair

It is mandatory for an ocean model to represent accurately the different kinds of waves since they play a critical role in ocean dynamics. Quantifying the dispersion or dissipation errors of a given numerical scheme and comparing numerical methods is not an easy task especially when using unstructured grids. In this paper we use a general method fully independent of the numerical scheme and of the grid to analyse dispersion and dissipation errors. In particular we apply this method to the study of the P1NC-P1 finite element pair applied to the shallow water equations. The influence of the grid is observed by comparing the convergence rates of the dispersion errors on Poincaré, Kelvin and Rossby waves. We observe a significative reduction of the convergence rate on unstructured meshes compared to structured grids for the P1NC-P1 pair, while this rate remains unchanged when using other approaches as the P1-P1P1-P1 pair without stabilization or the discontinuous Galerkin method.