Mathematical Sciences: Numerical Methods & Conservation Laws

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The investigator develops multi-dimensional high resolution finite volume methods for solving nonlinear hyperbolic systems of conservation laws and related problems arising in a variety of applications. The public domain software package CLAWPACK (Conservation LAWs PACKage) he has developed is extended to handle a wider variety of problems on both Cartesian and curvilinear grids in 1, 2, and 3 space dimensions. Mosaic composite grids are further developed to allow body-fitted grids near an irregular boundary to be coupled with Cartesian grids away from the boundary. Immersed interface methods for handling discontinuities in the solution or its derivatives are used in conjunction with multi-dimensional conservation law methods to solve fluid dynamics and wave propagation problems with material interfaces. These techniques achieve second order accuracy on uniform Cartesian grids cut by irregular interfaces. Similar techniques are applied to problems with stiff source terms arising from chemical reactions or combustion, giving rise to thin reaction zones that behave macroscopically as interfaces. Specific applications in a number of areas are studied, including groundwater flow, atmospheric flow, chemotaxis, and astrophysics. The software being developed is intended for teaching as well as research purposes, and includes extensive documentation and applied examples. An accompanying textbook is being written. The investigator develops computational methods and public domain software for the solution of a class of mathematical problems that arises in virtually every field of science and engineering. The partial differential equations considered can, in various forms, model the motion of liquids or gas (e.g., air in the atmosphere, water in the ocean, aerodynamic flow around aircraft or through turbines, groundwater or oil beneath the earth's surface), or the motion of waves in fluid or air (e.g., acoustic waves in the air or ocean or in ultrasonic explor ation of the body, seismic waves in the earth originating from earthquakes or artificially generated for oil exploration, radar waves). Even the motion of organisms in ecological modeling or cells in developmental biology follows similar laws. The methods are based on extensive research over the past 20 years, primarily in the aerodynamics and weapons development communities. This technology is slowly being transferred to other areas, but is hindered by the complexity of most of the algorithms. The software of this project should help speed this process. It is designed for general use as both a teaching and research tool, with extensive examples included in many applications areas. Novel methods are also developed to deal with phenomena occurring on different time scales (e.g., fast chemical reactions coupled with slow groundwater or atmospheric flow) and for problems in geometrically complicated regions of space bounded by irregular boundaries or containing interfaces where material properties change (e.g., between different types of rock in groundwater flow and seismology, or between bone and tissue in ultrasound imaging). Close collaboration is underway with researchers in many areas (particularly groundwater flow and atmospheric modeling), both to improve and generalize the software and to use it in the solution of specific problems. An application of great interest is contaminant transfer in groundwater flow, where linear or nonlinear advection in a porous medium with discontinuous permeabilities and irregular geometries must often be coupled with stiff source terms for adsorption and reactions. Accurate models are needed both as an aid to remediation of polluted sites and to the study of proposed underground storage sites for nuclear waste. Atmospheric modeling is crucial both in short-term weather prediction and in long-range global modeling of climate, ozone depletion, etc. The investigator works with researchers in these areas to incorporate this software into standard models as well as to develop new methods where needed.