Dispersion analysis of compatible Galerkin schemes for the 1D shallow water model

In this work, we study the dispersion properties of two compatible Galerkin schemes for the 1D linearized shallow water equations: the $P_n^C-P_{n-1}^{DG}$ and the $G{D}_n-DG{D}_{n-1}$ element pairs. $P_n$ is the order n Lagrange space, $P_{n-1}^{DG}$ is the order $n-1$ discontinuous Lagrange space, $G{D}_n$ is the order n Galerkin difference space, and $DG{D}_{n-1}$ is the order $n-1$ discontinuous Galerkin difference space. Compatible Galerkin methods have many desirable properties, including energy conservation, steady geostrophic modes and the absence of spurious stationary modes, such as pressure modes. However, this does not guarantee good wave dispersion properties. Previous work on the $P_2^C-P_1^{DG}$ pair has indeed indicated the presence of spectral gaps, and it is extended in this paper to the study of the $P_n^C-P_{n-1}^{DG}$ pair for arbitrary n. Additionally, an alternative element pair is introduced, the $G{D}_n-DG{D}_{n-1}$ pair, that is free of spectral gaps while benefiting from the desirable properties of compatible elements. Asymptotic convergence rates are established for both element pairs, including the use of inexact quadrature (which diagonalizes the velocity mass matrix) for the $P_n^C-P_{n-1}^{DG}$ pair and reduced quadrature for the $G{D}_n-DG{D}_{n-1}$ pair. Plots of the dispersion relationship and group velocities for a wide range of n and Rossby radii are shown. A brief investigation into the utility of mass lumping to remove the spectral gaps for the $P_3^C-P_2^{DG}$ pair is performed. Finally, a pair of numerical simulations are run to investigate the consequences of the spectral gaps and highlight the main differences between the two elements.