A number of physical problems involve localized phenomena that move in time, such as fluid fronts, interfaces and shocks. Often, these phenomena are not captured well by finite difference methods on uniform grids. The investigators work on three classes of such problems: flow in porous media, particularly chemically enhanced aquifer remediation; thin film flow with fluid fronts due to vanishing diffusive fluxes; and excitable media, particularly the modeling of the propagation of electrical impulses in the heart. All three of these problems involve diffusion; the diffusion can be nonlinear, subdiffusive or superdiffusive. The investigators develop efficient multilevel iterative methods and adaptive mesh refinement tools to solve these problems on parallel processor machines. The power of computers has dramatically increased since they were introduced a half century ago, and this improvement in performance continues at a breathtaking pace. Nonetheless, although the public is generally unaware of this, the computational resources required to simulate many phenomena, like the weather, greatly exceed those currently available. One way to obtain a lot of computer power is to arrange for many computers to work simultaneously on a problem. However, it is a daunting task to program these computers so that they will correctly produce the desired simulation in a timely fashion. In this work, the investigators develop effective methods to use many computers working together to simulate the flow of contaminants in the ground, the behavior of thin films, and the electrophysiology of the heart. Groundwater contamination is a significant environmental problem. Thin film problems arise in lubrication and in manufacturing processes. A fundamental understanding of the electrical and muscular behavior of the heart is important and could lead to better control of the mechanisms of arrhythmia.
