Finite-Volume Methods for Hyperbolic Problems

本建议涉及求解双曲型偏微分方程的多维高分辨率有限体积方法的发展，实现这些方法的软件的发展，以及这些方法在特定问题上的应用。这些方法在CLAWPACK软件包中实现，该软件包在网络上免费提供，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。这些算法和软件将被进一步发展，并被用于更广泛的问题。特别感兴趣的问题包括：基于新的树结构代码的自适应细化的进一步开发；包含笛卡尔网格技术的复杂几何；求解曲线流形上双曲型方程的一般方法的发展非均质材料中的波传播问题（如弹性动力学），包括具有空间变化通量函数的非线性问题。科学和工程中广泛的实际问题涉及波的传播或流体中物质的输运。这样的例子出现在各种各样的问题中，如研究人体组织中的超声波，地下水或大气中污染物的运输，以及研究黑洞碰撞产生的引力波。从数学上讲，所有这些问题都会导致相似的偏微分方程组。用数值方法解决这些方程需要特殊的技术来处理不连续函数，因为通常描述问题的系数或解（或两者）都是不连续的。例子包括超声问题中组织和骨骼界面处材料特性的不连续，或者在大多数非线性波传播问题中出现的冲击波。在过去的几十年里，已经发展了一类强大的数值方法来解决这类问题，这些问题在某些应用领域比其他应用领域使用得更多。该项目的一个主要目标是促进这项技术向新地区的转让。由P.I.及其同事开发的软件包CLAWPACK，旨在使学生学习这些方法和研究人员应用这些方法相对容易。在将这些方法应用于新情况时，仍然存在许多数学上的挑战。随着各种新应用的探索，将进行研究以进一步改进这些方法。本提案所涵盖的科目是计算数学研究生教育的沃土。P.I.通过接待来访的研究生，积极参与培训华盛顿大学和其他机构的学生和博士后。私家侦探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课堂讲稿、教科书、软件和其他教育材料。

This proposal concerns the development of multidimensional high-resolution finite-volume methods for solving hyperbolic partial differential equations, the development of software implementing these methods, and the application of these methods to particular problems. These methods are implemented in the CLAWPACK software package, which is freely available on the web and 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 will be further developed and brought to bear on a wider variety of problems. Particular problems of interest include: further development of adaptive refinement base on a new tree-structured code; inclusion of Cartesian-grid techniques for complex geometries; development of a general methodology for solving hyperbolic equations on curved manifolds; and wave-propagation problems (e.g., elastodynamics) in heterogeneous material, including nonlinear problems with spatially-varying flux functions.A wide range of practical problems in science and engineering involve the propagation of waves or the transport of substances in fluid flow. Examples arise in problems as diverse as the study of ultrasound waves in human tissue, the transport of contaminants in groundwater or the atmosphere, and the study of gravitational waves arising from the collision of black holes. Mathematically all of these problems lead to similar sets of partial differential equations. Solving these equations numerically requires special techniques that can deal with discontinuous functions, since often either the coefficients describing the problem or the solution (or both) are discontinuous. Examples include discontinuities in material properties at the interface between tissue and bone in an ultrasound problem, or the shock waves that arise in most nonlinear wave-propagation problems. Over the past few decades, a powerful class of numerical methods has been developed for solving such problems that have been much more heavily used in some applications areas than others. A primary goal of this project is to facilitate the transfer of this technology to new areas. The software package CLAWPACK, developed by the P.I. and coworkers, is designed to make it relatively easy to for students to learn about these methods and for researchers to apply them. There are still numerous mathematical challenges that arise in applying these methods to new situations. Research will be conducted to further improve these methods as a variety of new applications are explored. The subjects covered in this proposal are fertile ground for graduate education in computational mathematics. The P.I. is actively involved in training students and postdocs at the University of Washington as well as at other institutions by hosting visiting graduate students. The P.I. has also taught several short courses elsewhere and developed lecture notes, textbooks, software, and other educational material based on this research.