A discontinuous Galerkin method for two- and three-dimensional shallow-water equations

The shallow-water equations (SWE), derived from the incompressible Navier-Stokes equations using the hydrostatic assumption and the Boussinesq approximation, are a standard mathematical representation valid for most types of flow encountered in coastal sea, river, and ocean modeling. They can be utilized to predict storm surges, tsunamis, floods, and, augmented by additional equations (e.g., transport, reaction), to model oil slicks, contaminant plume propagation, temperature and salinity transport, among other problems. An analytical solution of these equations is possible for only a handful of particular cases. However, their numerical solution is made challenging by a number of factors. The SWE are a system of coupled nonlinear partial differential equations defined on complex physical domains arising, for example, from irregular land boundaries. The bottom sea bed (bathymetry) is also often very irregular. Shallow-water systems are subjected to a wide range of external forces, such as the Coriolis force, surface wind stress, atmospheric pressure gradient, and tidal potential forces. As a result, flow regimes can vary greatly throughout the domain, from very smooth to high gradients and shock waves. The solution of the system is further complicated by the difficulties connected with the mathematical nature of the SWE. Most important is the coupling between the gravity forcing and the horizontal velocity field, which could lead to spurious spatial oscillations if the numerical algorithms are not chosen with care. One has to note, though, that most existing numerical methods for the SWE have serious drawbacks with regard to stability, local conservation, and ability to accommodate parallel implementation and hp-adaptivity. These problems become even more evident if we try to simulate problems involving discontinuities, shock waves, etc. The discontinuous Galerkin (DG) methods are an attempt to marry the most favorable features of the continuous finite element and finite volume schemes. On the one hand, they can employ approximating spaces of any order (not necessarily polynomial), and, on the other, the numerical fluxes on the inter-element boundaries are evaluated exactly as in the finite volume method – by solving a Riemann problem. As a result, these numerical schemes enjoy the same stability properties as the finite volume method. In addition, most DG methods guarantee local conservation of mass and momentum, which is, in many cases, a highly desirable quality reflecting the physical nature of the processes we are trying to model. In this thesis, we formulate the local discontinuous Galerkin method for the 2- and 3D shallow-water equations and derive stability and a priori error estimates for a simplified form of the 2D shallow-water equations and conduct stability analysis of our 3D scheme. In a series of numerical studies, we test both formulations using problems with discontinuous solutions as well as typical tidal flow problems. In addition, we demonstrate adaptive capabilities of the method using a shock-detection algorithm as an error indicator.Computational Science, Engineering, and MathematicsComputational and Applied Mathematic