Finite Volume Methods and Software for Hyperbolic Problems

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Nonlinear hyperbolic systems of partial differential equations arise in many scientific and engineering applications where wave propagation or transport phenomena are important, giving rise to discontinuous solutions such as shock waves. Often the problems are posed in heterogeneous media in which the material parameters vary and may be discontinuous across interfaces. Solving these equations requires special techniques based on the mathematical theory of weak solutions. The investigator has previously developed a multidimensional "wave-propagation algorithm" that yields a very general framework for solving such problems. These high-resolution finite volume methods are implemented in the open source Clawpack software package, which allows students and researchers studying a wide range of phenomena to use the technology of high-resolution methods and adaptive mesh refinement. These algorithms and the software are being further developed and brought to bear on a variety of problems, particularly in geophysical flow and wave propagation, in collaboration with researchers in these fields. A version of the Clawpack software (GeoClaw) specifically tuned to model tsunami propagation and innundation has recently been developed and is being extended to handle various related problems involving flow over topography. The study of specific applications is motivating the development of new mathematical models and finite volume algorithms to more accurately and robustly model these extreme flows.This work focuses on the development of computational models for simulating geophysical flow and wave propagation problems that are related to the prediction and mitigation of natural disasters. The investigator and coworkers been working on tsunami modeling for several years and have developed a software package that has been used both for scientific research on tsunami phenomena and for the preparation of innundation maps for specific communities in the Pacific Northwest. In addition to continuing this work, the software is being extended to model other related phenomena, which often requires the development of new mathematical models and computational algorithms. Applications being studied include: flooding due to catastrophic dam breaks, storm surges generated by hurricanes such as Katrina, tsunamis generated by submarine landslides, and the flooding and debris flows that would result from the eruption of volcanoes such as Mt. Rainier that are covered with glaciers. The investigator and his students are working with researchers in earth sciences, at the USGS Cascades Volcano Observatory, and at several other research centers on scientific and hazard mitigation projects related to this software development. New algorithms for modeling earthquakes are also being developed and applied to the study of tremors at Mount St. Helens in order to help determine the risk of future eruptions. The investigator is actively involved in training students and postdoctoral fellows at the University of Washington as well as at other institutions by hosting visiting graduate students and scientists. The investigator has also taught several short courses elsewhere and has developed lecture notes, textbooks, software, and other educational material based on this research. The software enhancements under way will further improve its usability as a freely-available educational tool for students and researchers in many fields where similar problems arise.