Application of Distributed Adaptive Mesh Refinement Techniques to Problems Involving Diffusion

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A number of physical problems involve localized phenomenathat move in time, such as fluid fronts, interfaces and shocks.Often, these phenomena are not captured well by finite differencemethods on uniform grids. The investigators work on threeclasses of such problems: flow in porous media, particularlychemically enhanced aquifer remediation; thin film flow withfluid fronts due to vanishing diffusive fluxes; and excitablemedia, particularly the modeling of the propagation of electricalimpulses in the heart. All three of these problems involvediffusion; the diffusion can be nonlinear, subdiffusive orsuperdiffusive. The investigators develop efficient multileveliterative methods and adaptive mesh refinement tools to solvethese problems on parallel processor machines. The power of computers has dramatically increased sincethey were introduced a half century ago, and this improvement inperformance continues at a breathtaking pace. Nonetheless,although the public is generally unaware of this, thecomputational resources required to simulate many phenomena, likethe weather, greatly exceed those currently available. One wayto obtain a lot of computer power is to arrange for manycomputers to work simultaneously on a problem. However, it is adaunting task to program these computers so that they willcorrectly produce the desired simulation in a timely fashion. Inthis work, the investigators develop effective methods to usemany computers working together to simulate the flow ofcontaminants in the ground, the behavior of thin films, and theelectrophysiology of the heart. Groundwater contamination is asignificant environmental problem. Thin film problems arise inlubrication and in manufacturing processes. A fundamentalunderstanding of the electrical and muscular behavior of theheart is important and could lead to better control of themechanisms of arrhythmia.