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.

双曲型偏微分方程组在许多重要的波传播或传输现象的应用中出现，这些方程和/或它们的解往往涉及间断函数，这给标准的有限差分方法离散微分方程组带来了困难。特别是，非线性波传播问题经常引起激波，即使从光滑的初始数据也可以自发地在解中产生不连续。然后目标是近似基本积分守恒律的弱解。通常，这个问题必须在非均匀介质中解决，在这种介质中，材料的性质随空间而变化，通常在尖锐的材料界面上不连续。这导致要求解的方程中的系数或通量函数不连续。这项建议涉及进一步发展多维高分辨率有限体积法来解决这类问题，开发实现这些方法的软件，并将这些方法应用于特定的问题。P.I.以前开发了一种多维“波传播算法”，为解决这类问题提供了一个非常通用的框架，并在CLAWPACK软件中实现了这种方法。这些算法和软件将被进一步开发和应用于各种问题。需要研究的一些特殊应用包括：海啸传播和上升、火山喷发产生的火山碎屑流、地震波的模拟以及弹性波在非均匀介质中的传播，包括冲击波在组织和骨骼中的传播。许多科学和工程中出现的实际问题都是用“双曲微分方程”模拟的，具有非常相似的数学结构，使应用数学和计算数学的研究人员做出了广泛适用的贡献。这项工作的目标是进一步开发方法和软件来近似这些方程的解。这些方法是在由P.I.和同事编写的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最先进的方法来解决这些数学问题。在过去的几年里，这个软件已经被5000多个注册用户下载，并被PI和他的学生以及其他用户应用了大量的科学和工程问题。具体的实际问题也将在P.I.和学生已经完成的工作的基础上进行研究。其中一个项目涉及对海啸对沿海地区的影响进行建模，既是为了帮助对过去的海啸进行科学研究，也是为了帮助减灾和备灾。其他地球物理项目包括研究火山喷发产生的流动和地震后地震波在地球上的传播或在石油勘探中。一个具有生物医学应用的项目是研究冲击波在组织和骨骼中的传播，潜在地应用于“冲击波疗法”的研究，在“冲击波疗法”的研究中，超声波冲击波被用于治疗各种医疗条件，包括骨不连(骨折无法愈合)、足底筋膜炎和肌腱炎。