Efficiency Improvements for Modeling Coastal Hydrodynamics through the Application of High-Order Discontinuous Galerkin Solutions to the Shallow Water Equations

Storm surge is the rise of water caused by hurricane winds blowing over large regions of the ocean surface. This rise of water induces coastal flooding and is responsible for much of the destruction of property and loss of life caused by hurricane events. As computational resources have grown, numerical simulation has become a powerful tool in the mitigation of these disasters. Today's state-of-the-art surge models, based on relatively robust low-order numerical methods, have become increasingly accurate with the inclusion of additional physical processes and the use of high-resolution domains. However, these improvements have come at the expense of rising computational cost, demanding large high-performance computing resources. As a result, some analyses must resort to older, less advanced models that sacrifice accuracy for the sake of lower computational effort.

The goal of this dissertation is to increase the efficiency of high-fidelity coastal modeling through the application of high-order numerical methods. Although high-order methods promise greatly increased efficiency, they have not yet been widely adopted in the coastal surge modeling community. While inherently more accurate than low-order methods, high-order schemes are also much less diffusive, often leading to stability issues for complex applications. Additionally, the computational meshes and parameter sets that describe a given model domain must incorporate high-order representations of the underlying data in order to maintain consistency with the accuracy of the numerical method. Techniques for accomplishing this in real-world coastal circulation problems are not well developed.

In addition to replacing low-order algorithms, many fundamental aspects of traditional surge modeling techniques must be reconsidered in order to fully realize the potential performance improvements of high-order methods for real-world problems. Simply incorporating high-order methods into today's models is not a viable solution. Despite their greater accuracy, they are also more expensive than low-order methods at a given resolution level. As a result, efficiency gains must come from using high-order solution approximations along with coarser discretizations to obtain a level of accuracy that would require higher resolution for a low-order approach. This means boundary conditions and input parameter fields (i.e. bathymetric depth, wind stress, Manning's n friction, etc.) that are typically comprised of high-resolution linear approximations, must now be resolved through higher-order polynomials inside larger elements. This dissertation seeks to apply high-order discontinuous Galerkin finite element solutions of the shallow water equations to realistic coastal tidal and surge problems.

The goal is to obtain results on par with the accuracy of current low-order models, while significantly surpassing their performance. Accomplishing this goal requires developing boundary conditions, parameter field representations, and mesh discretizations that are consistent with the high-order accuracy of the numerical methods. In addition, coding paradigms can be adapted to exploit current computational technologies to further improve performance. These developments will allow coarse, high-order models to provide comparable accuracy to today's state-of-the-art high-resolution, low-order models at greatly reduced computational expense.