Dispersion analysis of compatible Galerkin schemes on quadrilaterals for shallow water models

International audienceCompatible Galerkin methods (the Galerkin analogue of an Arakawa C-Grid) are growing in popularity for simulating geophysical fluid flows, due to their desirable characteristics, including but not limited to: energy conservation, higher-order accuracy, steady geostrophic modes and the absence of spurious stationary modes, such as pressure modes. However, these characteristics still do not guarantee good wave dispersion properties. In this work, we study the dispersion properties of two compatible Galerkin families for the 2D linear rotating shallow water equations on quadrilaterals: the $Q_n^-\Lambda^k$ family from finite element exterior calculus and the newly developed $MGD_n$ family. These families are the extensions to quadrilaterals of the $P_n^C-P_{n-1}^{DG}$ and $GD_n-DGD_{n-1}$ pairs, respectively, studied for the 1D linear shallow water equations in [13]. A major finding from that paper was that the $P_n^C-P_{n-1}^{DG}$ pair has spectral gaps for $n\geq2$ and the $GD_n-DGD_{n-1}$ does not. These spectral gaps are non-dimensional wavenumbers where the dispersion relationship is double-valued, and lead to anomalous dispersion and noise in numerical simulations. On quadrilaterals, previous work [24, 26] on the $Q_n^-\Lambda^k$ family for inertia waves with $n=2$ and for gravity waves for arbitrary n has indicated the presence of spectral gaps, in the form of line discontinuities. The investigation of these gaps for the $Q_n^-\Lambda^k$ family is extended in this paper to inertia-gravity waves for arbitrary $n$, including plots of the dispersion relationship for $n=2$ when using the lumping developed in [26, 35] that eliminates the spectral gaps. Additionally, the $MGD_n$ family is studied (including the use of reduced quadrature), which is found to be free of spectral gaps. For both families asymptotic convergence rates are established, effective resolutions determined and plots of the dispersion relationships for a range of $n$ and Rossby radii are shown. Finally, a pair of numerical simulations are run to investigate the consequences of the spectral gaps and highlight the main differences between the two families