Higher Order Tsunami Simulations and Novel Solution Methods

Often caused with little to no warning, tsunamis can cause devastating damage to coastal cities and island communities. When a tsunami occurs, it is imperative for first responders and public safety officials to have accurate information about where the tsunami will make landfall and how big the waves are projected to be. Tsunamis can be modeled using a set of equations called the Shallow Water Equations, derived from the Navier-Stokes equations for fluid momentum. A higher order model called the Dispersive Shallow Water Equations can be formulated with a higher order analysis from the Navier-Stokes equations. Although this higher order model is more accurate and captures better physics, it is more computationally expensive to work with. To efficiently solve the Dispersive Shallow Water Equations, we are implementing a novel partial differential equation (PDE) solver called the Hierarchical Poincaré-Steklov (HPS) method. The HPS method is a direct solver for elliptic PDEs that works by building a global solution operator by recursively merging small sections of the grid called patches. We are implementing the HPS method to work with a technique called adaptive mesh refinement (AMR) to allow the user to solve with high resolution where needed, and low resolution where not needed. We show that our current implementation is comparable in speed and accuracy to the highly efficient FISHPACK solver for elliptic PDEs. We are currently working on implementing the HPS method in parallel on CPUs and GPUs to run on supercomputer architectures