Finite-Volume Methods for Hyperbolic Problems

这项建议涉及到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开发，以及这些方法在特定问题中的应用。这些方法是在CLAWPACK软件包中实现的，该软件包可以在网络上免费获得，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。这些算法和软件将被进一步开发，并用于更广泛的问题。特别感兴趣的问题包括：基于新的树形结构代码的自适应精化的进一步发展；复杂几何的笛卡尔网格技术；在曲线流形上求解双曲型方程的一般方法的发展；以及非均匀材料中的波传播问题(例如，弹性动力学)，包括具有空间变化的通量函数的非线性问题。科学和工程中的广泛的实际问题涉及波的传播或物质在流体中的传输。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，污染物在地下水或大气中的传输，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类似的偏微分方程组。数值求解这些方程需要能够处理不连续函数的特殊技术，因为描述问题或解(或两者)的系数通常是不连续的。例如，超声波问题中组织和骨骼界面的材料性质不连续，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，已经发展出一类强大的数值方法来解决这类问题，这些问题在一些应用领域比在另一些领域得到了更广泛的应用。该项目的一个主要目标是促进这项技术向新领域的转让。由P.I.和同事开发的CLAWPACK软件包旨在让学生相对容易地了解这些方法，并让研究人员应用它们。在将这些方法应用于新的情况时，仍然会出现许多数学挑战。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。这项建议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，积极参与了华盛顿大学和其他机构的学生和博士后培训工作。P.I.还在其他地方教授了几门短期课程，并在这项研究的基础上开发了课堂讲稿、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.