Finite Volume Methods and Software for Hyperbolic Problems

双曲问题的有限体积方法和软件

• 批准号: 1216732
• 负责人: Randall LeVeque
• 金额: $ 39.98万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1216732&HistoricalAwards=false
• 关键词: Finite; Volume; Methods; Software; Hyperbolic

This proposal will fund research on finite volume methods for solving linear and nonlinear hyperbolic systems of partial differential equations, and the continued development of the open source software package Clawpack. This software provides an implementation of a class of wave propagation algorithms that has been extensively tested and used by students and researchers in many fields, and also incorporates adaptive mesh refinement (AMR). Specific goals of this research include: (1) Development of Discontinuous Galerkin (DG) methods in the Clawpack framework, which could potentially provide higher order of accuracy for a certain class of problems and greater efficiency in spite of their higher cost per time step. (2) Solution of three-dimensional orthotropic poroelastic wave propagation problems, with applications to the continuing investigation of Extracorporeal Shock Wave Therapy (ESWT). (3) Development of adjoint equation technology for AMR error estimation and sensitivity analysis, with application to tsunami modeling in particular. (4) Development of multilevel Monte-Carlo capabilities and exploration of this and other uncertainty quantification techniques, with application to the probabilistic assessment of natural hazards.The class of algorithms being investigated can be used to solve a wide range of practical problems in science and engineering that involve wave motion. These problems are modeled by partial differential equations that cannot be solved exactly, and so numerical approximations must be generated computationally at millions of grid points. Development of better algorithms for efficiently computing accurate solutions is a primary goal of this research. Much of the work is quite general and can be applied to many different problems, and the algorithms are implemented in an open source software package Clawpack that is freely available at www.clawpack.org. Work supported by this grant is strongly motivated by two specific applications of practical interest. One is the simulation of hazardous geophysical flows, particularly tsunamis, and the development of techniques to more rapidly and accurately predict the effects of a tsunami. For risk assessment and hazard mitigation, it is also necessary to consider a range of potential earthquakes that could cause tsunamis in the future. Mathematical and computational techniques will be investigated that can aid in efficiently estimating the probability of inundation from a large number of potential tsunamis at many different points in a coastal community. The other motivating application is the study of shock wave propagation in tissue and bone and the mechanical stress generated that can lead to biological response, such as bone growth or wound healing. These studies are important in gaining a better understanding of shock wave therapy, a clinical procedure that has exhibited some significant results.

这项建议将用于研究解线性和非线性双曲型偏微分方程组的有限体积法，以及继续开发开放源码软件包Clawpack。该软件提供了一类已经被学生和研究人员在许多领域广泛测试和使用的波传播算法的实现，并结合了自适应网格加密(AMR)。这项研究的具体目标包括：(1)在Clawpack框架中开发不连续Galerkin(DG)方法，尽管其每一时间步的成本较高，但该方法可能对某类问题提供更高阶的精度和更高的效率。(2)三维正交异性多孔弹性波传播问题的求解，及其在体外冲击波治疗(ESWT)继续研究中的应用。(3)发展了AMR误差估计和灵敏度分析的伴随方程技术，特别是在海啸模拟中的应用。(4)发展多层次蒙特卡罗能力，探索这种和其他不确定性量化技术，并将其应用于自然灾害的概率评估。正在研究的这类算法可用于解决涉及波动的科学和工程中的广泛实际问题。这些问题都是用不能精确求解的偏微分方程组来模拟的，因此必须在数百万个网格点上通过计算产生数值近似。开发更好的算法来有效地计算准确的解是本研究的主要目标。许多工作是非常通用的，可以应用于许多不同的问题，算法是在一个开源软件包Clawpack中实现的，该软件包可以在www.clawpack上免费获得。这笔赠款支持的工作受到两项具有实际意义的具体应用的强烈推动。一个是模拟危险的地球物理流动，特别是海啸，以及开发更快、更准确地预测海啸影响的技术。为了进行风险评估和减轻灾害，还有必要考虑未来可能引发海啸的一系列潜在地震。将研究数学和计算技术，以帮助有效地估计沿海社区许多不同地点的大量潜在海啸造成淹没的可能性。另一个激励性的应用是研究冲击波在组织和骨骼中的传播以及产生的机械应力，这些应力可以导致生物反应，如骨骼生长或伤口愈合。这些研究对于更好地理解冲击波疗法是重要的，冲击波疗法是一种临床程序，已经展示了一些重要的结果。