Finite Volume Methods for Hyperbolic Problems

双曲型偏微分方程组出现在波的传播或输运现象很重要的许多应用中，这些方程和/或它们的解往往涉及到不连续函数，这给标准的有限差分方法离散微分方程带来了困难。 特别是，非线性波传播问题往往会产生激波，解中的不连续性甚至可以从光滑的初始数据中自发产生。 通常，这个问题必须在非均匀介质中解决，在这种介质中，材料特性随空间而变化，在尖锐的材料界面处经常是不连续的。 这将导致方程中的不连续系数或通量函数。 本建议涉及解决此类问题的多维高分辨率有限体积法的进一步发展，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。 私家侦探先前已经开发了一个多维的“波传播算法”，产生了一个非常通用的框架来解决这些问题，并在CLAWPACK软件中实现了这种方法。 这些算法和软件将进一步发展，并带来承担各种问题。一些特殊的应用进行研究包括：海啸的传播和爬升，火山爆发引起的火山碎屑流，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。科学和工程中出现的广泛的实际问题都是用“双曲微分方程”，并有一个非常相似的数学结构，使研究人员在应用和计算数学作出贡献，是广泛适用的。 这项工作的目标是进一步开发的方法和软件，这些方程的近似解。 这些方法都是在P.I.编写的CLAWPACK软件包中实现的。和同事，这是免费提供的网络，并允许学生和研究人员研究广泛的现象，使用国家的最先进的方法为这些数学问题。 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，他的学生和其他用户应用于大量的科学和工程问题。 具体的实际问题也将研究，建立在已经完成的工作由PI。和学生 其中一个项目涉及建立海啸对沿海地区影响的模型，既有助于对过去海啸的科学研究，也有助于减灾和备灾。 其他的地球物理项目包括研究火山爆发引起的流动和地震后地震波在地球中的传播或石油勘探。 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，并可能应用于“冲击波疗法”的研究，其中超声冲击波用于治疗各种医疗状况，包括骨不连（骨折无法愈合），足底筋膜炎和肌腱炎。

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.