Finite Volume Methods and Software for Hyperbolic Problems

偏微分方程的非线性双曲型系统出现在许多科学和工程应用中，在这些应用中，波的传播或输运现象是很重要的，它产生了不连续的解，如激波。通常问题是在非均质介质中提出的，其中材料参数变化，并且可能在界面上不连续。求解这些方程需要基于弱解数学理论的特殊技术。这位研究者之前开发了一种多维“波传播算法”，为解决这类问题提供了一个非常通用的框架。这些高分辨率有限体积方法是在开源的Clawpack软件包中实现的，它允许学生和研究人员研究广泛的现象，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发展，并与这些领域的研究人员合作，用于解决各种问题，特别是在地球物理流动和波传播方面。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，并正在扩展到处理涉及地形流动的各种相关问题。对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，以更准确和健壮地模拟这些极端流动。这项工作的重点是开发用于模拟与预测和减轻自然灾害有关的地球物理流动和波传播问题的计算模型。调查员及其同事几年来一直在研究海啸模型，并开发了一个软件包，用于海啸现象的科学研究和为太平洋西北地区的特定社区编制洪水地图。除了继续这项工作之外，该软件还被扩展到其他相关现象的建模，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：灾难性大坝溃坝引起的洪水、卡特里娜等飓风引起的风暴潮、海底滑坡引起的海啸，以及雷尼尔山等被冰川覆盖的火山爆发引起的洪水和泥石流。研究者和他的学生们正在与美国地质勘探局喀斯喀特火山观测站的地球科学研究人员以及其他几个与该软件开发相关的科学和减灾项目研究中心合作。地震模拟的新算法也正在开发中，并应用于圣海伦斯火山的地震研究，以帮助确定未来爆发的风险。通过接待来访的研究生和科学家，积极参与培训华盛顿大学和其他机构的学生和博士后。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。正在进行的软件增强将进一步提高其可用性，使其成为许多领域中出现类似问题的学生和研究人员的免费教育工具。

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.