Finite Volume Methods and Software for Hyperbolic Problems

非线性双曲型偏微分方程组出现在许多科学和工程应用中，其中波的传播或传输现象是重要的，从而产生不连续的解决方案，如冲击波。 这些问题通常出现在非均匀介质中，其中材料参数变化，并且在界面上可能是不连续的。 求解这些方程需要基于弱解数学理论的特殊技巧。 研究人员以前开发了一个多维的“波传播算法”，产生一个非常一般的框架来解决这些问题。 这些高分辨率有限体积方法在开源Clawpack软件包中实现，该软件包允许研究各种现象的学生和研究人员使用高分辨率方法和自适应网格细化技术。 这些算法和软件正在进一步开发，并与这些领域的研究人员合作，用于解决各种问题，特别是地球物理流动和波传播方面的问题。 最近开发了一个专门用于模拟海啸传播和淹没的Clawpack软件（GeoClaw）版本，并将其扩展到处理涉及地形流动的各种相关问题。 具体应用的研究正在推动新的数学模型和有限体积算法的发展，以更准确和鲁棒地模拟这些极端的流。这项工作的重点是开发计算模型，模拟地球物理流和波传播问题，与预测和减轻自然灾害。 研究人员和同事们多年来一直致力于海啸建模，并开发了一个软件包，用于海啸现象的科学研究和太平洋西北部特定社区的淹没地图的编制。除了继续这项工作外，该软件正在扩展到模拟其他相关现象，这通常需要开发新的数学模型和计算算法。 正在研究的应用包括：由于灾难性的大坝破裂造成的洪水，卡特里娜飓风等飓风产生的风暴潮，海底滑坡产生的海啸，以及火山爆发造成的洪水和泥石流，如Mt.它们被冰川覆盖。 研究人员和他的学生正在与地球科学研究人员合作，在USGS瀑布火山天文台，并在其他几个研究中心的科学和减灾项目与此软件开发。 还正在开发模拟地震的新算法，并将其应用于研究圣海伦斯山的震颤，以帮助确定未来爆发的风险。 研究员积极参与培训华盛顿大学以及其他机构的学生和博士后研究员，接待访问研究生和科学家。 研究者还在其他地方教授了几门短期课程，并根据这项研究开发了讲义、教科书、软件和其他教育材料。 正在进行的软件改进将进一步提高其可用性，使其成为出现类似问题的许多领域的学生和研究人员可以免费使用的教育工具。

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.