Finite Volume Methods for Hyperbolic Problems

双曲问题的有限体积方法

• 批准号: 0609661
• 负责人: Randall LeVeque
• 金额: $ 29.99万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-09-01 至 2010-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0609661&HistoricalAwards=false
• 关键词: Finite; Volume; Methods; Hyperbolic; Problems

Hyperbolic systems of partial differential equations arise in manyapplications where wave propagation or transport phenomena are important.Often these equations and/or their solutions involve discontinuousfunctions, giving difficulties for standard finite-difference approachesto discretizing the differential equations. In particular, nonlinear wavepropagation problems often give rise to shock waves, discontinuities inthe solution which can arise spontaneously even from smooth initial data.The goal is then to approximate a weak solution to the underlying integralconservation law. Often the problem must be solved in a heterogeneousmedium where the material properties vary with space, often discontinouslyat sharp material interfaces. This results in discontinuous coefficientsor flux functions in the equations to be solved. This proposal concernsthe further development of multidimensional high-resolutionfinite volume methods for solving such problems, the development ofsoftware implementing these methods, and the application of thesemethods to particular problems. The P.I. has previously developeda multidimensional "wave-propagation algorithm" that yields a verygeneral framework for solving such problems, and has implemented thismethod in the CLAWPACK software. These algorithms and the softwarewill be further developed and brought to bear on a variety of problems.Some particular applications to be studied include: tsunami propagationand runup, pyroclastic flows arising from volcanic eruptions, thesimulation of seismic waves, and elastic wave propagation in heterogeneousmedia, including shock wave propagation in tissue and bone.A wide range of practical problems arising in science and engineering aremodeled using "hyperbolic differential equations" and have a very similarmathematical structure, allowing researchers in applied and computationalmathematics to make contributions that are widely applicable. The goalof this work is to further develop methods and software for approximatingthe solutions to these equations. These methods are implemented in theCLAWPACK software package written by the P.I. and co-workers, whichis freely available on the web and allows students and researchersstudying a wide range of phenomena to use state of the art methods forthese mathematical problems. This software has been downloaded by morethan 5000 registered users over the past several years and applied tonumerous scientific and engineering problems by the PI, his students, andother users. Specific practical problems will also be studied, building onwork already performed by the P.I. and students. One project involvesmodeling the effects of tsunamis on coastal regions, both to aid inscientific studies of past tsunamis and as an aid to hazard mitigationand preparedness. Other geophysical projects involve the study of flowsarising from volcanic eruptions and the propagation of seismic waves inthe earth following an earthquake or in oil exploration. A project withbiomedical applications is the study of shock waves propagating in tissueand bone, with potential application to the study of "shock wave therapy",in which ultrasonic shock waves are used to treat a variety of medicalconditions including nonunions (broken bones that fail to heal), plantarfasciitis, and tendinitis.

偏微分方程的双曲系统出现在波传播或传输现象很重要的许多应用中。这些方程和/或其解通常涉及不连续函数，这给标准有限差分方法离散微分方程带来了困难。 特别是，非线性波传播问题经常会产生冲击波，甚至在平滑的初始数据中也会自发地产生解的不连续性。目标是逼近基本积分守恒定律的弱解。 通常，该问题必须在异质介质中解决，其中材料特性随空间变化，通常在尖锐的材料界面处不连续。 这导致待解方程中的系数或通量函数不连续。 该提案涉及解决此类问题的多维高分辨率有限体积方法的进一步开发、实现这些方法的软件的开发以及这些方法在特定问题上的应用。 P.I.先前开发了一种多维“波传播算法”，该算法产生了解决此类问题的非常通用的框架，并在 CLAWPACK 软件中实现了该方法。 这些算法和软件将进一步开发并应用于各种问题。要研究的一些特定应用包括：海啸传播和上升、火山喷发引起的火山碎屑流、地震波模拟以及非均质介质中的弹性波传播，包括组织和骨骼中的冲击波传播。科学和工程中出现的各种实际问题都使用“双曲微分方程”进行建模，并且具有非常相似的数学结构，使研究人员能够在应用和工程中进行研究。计算数学做出了广泛适用的贡献。 这项工作的目标是进一步开发用于逼近这些方程解的方法和软件。 这些方法在 P.I. 编写的 CLAWPACK 软件包中实现。和同事，它可以在网络上免费获得，并允许研究各种现象的学生和研究人员使用最先进的方法来解决这些数学问题。 在过去的几年里，该软件已被超过 5000 名注册用户下载，并由 PI、他的学生和其他用户解决了许多科学和工程问题。 还将在 P.I. 已经完成的工作的基础上研究具体的实际问题。和学生。 其中一个项目涉及模拟海啸对沿海地区的影响，既可以帮助对过去海啸进行非科学研究，也可以帮助减轻灾害和做好准备。 其他地球物理项目涉及研究火山喷发产生的流动以及地震后或石油勘探中地震波在地球中的传播。 生物医学应用项目是研究冲击波在组织和骨骼中传播，可能应用于“冲击波疗法”的研究，其中超声波冲击波用于治疗各种医疗状况，包括骨不连（无法愈合的骨折） ）、足底筋膜炎和肌腱炎。