In this article, we present novel, high-order, discontinuous Galerkin (DG) methods for the vertical extent of the water column in coastal settings. We examine the shallow water equations (SWE) in the context of DG spatial discretizations coupled with explicit Runge-Kutta (RK) time stepping. All the primary variables, including the free surface elevation, are discretized using discontinuous polynomial spaces of arbitrary order. The difficulty of mismatched lateral boundary faces that accompanies the use of a discontinuous free surface is overcome through the use of a so-called sigma-coordinate system in the vertical, which transforms the bottom boundary and free surface into coordinate surfaces. We develop high-order methods for the SWE that exhibit optimal orders of convergence for all the primary variables via two distinct paths: the first involves the use of a convolution kernel made up of B-splines to filter out errors in the DG discretization of the surface elevation and the corresponding pressure flux. The second involves a method that evaluates the discrete depth-integrated velocity exactly, eliminating the need to solve the depth-integrated momentum equation altogether. The result is a simple and efficient high-order scheme that can be extended to the full three-dimensional SWE.
Conroy, C. J., & Kubatko, E. J. (2016). hp discontinuous Galerkin methods for the vertical extent of the water column in coastal settings part I: Barotropic forcing. Journal of Computational Physics, 305, 1147–1171. https://doi.org/10.1016/j.jcp.2015.10.038