Space-time hybridized discontinuous Galerkin methods for shallow water equations
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The non-linear shallow water equations model the dynamics of a shallow layer of an incompressible fluid; they are obtained by asymptotic analysis and depth-averaging of the Navier-Stokes equations. They are utilized in a wide range of applications, from simulation of geophysical phenomena such as river/oceanic flows and avalanches to the study of hurricane simulation, storm surge modeling, and oil spills. As a hyperbolic system of equations, shocks may develop in finite time and therefore an appropriate numerical discretization of these equations needs to be developed. The purpose of this dissertation is to develop and implement a state of the art numerical method to accurately model these equations. Therefore, a well-balanced space-time hybridized discontinuous Galerkin method was developed for our purpose. The method was implemented and tested for several benchmark problems and very promising results were obtained. An a priori error estimate for the developed method was also obtained with an optimal rate of convergence in an appropriate norm. The estimate obtained is an extension of the existing a priori error estimates in the literature, first to the case of a system of shallow water equations, second to a hybridized mixed DG method, and third to an arbitrary degree of polynomial in time.