An h-box Method for Shallow Water Equations

The model equations for storm surge and tsunamis most commonly used are the shallow water equations with addition of appropriate source terms for bathymetry. Traditional approaches will need to resolve the mesh to discretize small-scale structure, which impacts the time-step size to be proportional to the size of cells. In this thesis, a novel approximate Riemann solver was developed in order to deal with the existence of barrier without restricting the time-step due to small cells. Because of the wave redistribution method and proper ghost cells setting, the novel Riemann solver maintained properties including mass and momentum conservation, the well-balancing properties and robustness at the wet-dry interface. The solver also preserves nonnegative water depth and prevents leakage. A modified h-box method is applied so the algorithm can overcome restrictions of small time-step sizes. The work has been done in the context of the GeoClaw platform with retaining the capabilities of GeoClaw solver. At the same time, the special developed Riemann solver extends the package to handle the sub-grid-scale effects of barriers. Incorporating the solver developed in this work into the GeoClaw framework has allowed to leverage GeoClaw’s ability to handle complex bathymetry and real applications.