Development of Cut Cell Methods for Barrier Simulations with Shallow Water Equations

In this thesis we aim to provide computationally efficient methods of performing waterbarrier simulations. The innate challenge in simulations of structures such as sea or surge barriers is resolution. Because barriers tend to be long and thin compared to the surrounding landscapes they protect, one must put mesh refinement on the barrier region in order to even numerically recognize the barrier’s presence. This is a costly computation due to the CFL condition which puts a strict limit on the size of time step proportional to the spatial mesh size. Another issue is the complexity of meshing near the barrier. Since barriers are most likely slanted or have certain shapes, the grid has to reflect this in the form of a grid mapping or an unstructured grid. To mitigate the issue of resolution, we propose an approximation of the barrier with a line interface embedded on a Cartesian grid, reducing our problem to an embedded boundary problem. Then to avoid complex meshing, we develop three cut cell methods on two shapes of barriers: 1) the h-box method (HB), 2) the state redistribution method (SRD), and 3) the cell merging method (CM). Doing this two-step approach means that we can lower the resolution near the barrier region and still feel the presence of the barrier and capture its effect, which would otherwise not be the case if we relied on resolution for representation of the barrier. This does not mean that we are losing accuracy by lowering resolution, however. Rather, we are maintaining about the same accuracy while also lowering resolution (and thus cutting computational cost), which we show by comparison with a refined barrier. We solve the shallow water equations as our underlying PDEs to simulate water interaction with the barrier, as they are commonly used in tsunami and storm simulations. We implement our work on the PYCLAW framework, which is an objected oriented program that solves conservation laws.