Finite Volume Methods and Software for Hyperbolic Problems

该提案将资助求解线性和非线性双曲型偏微分方程组的有限体积方法的研究，以及开源软件包Clawpack的持续开发。该软件提供了一类波传播算法的实现，该算法已被许多领域的学生和研究人员广泛测试和使用，并且还包含自适应网格细化（AMR）。本研究的具体目标包括：(1)在Clawpack框架下开发不连续Galerkin （DG）方法，尽管其每时间步的成本较高，但可能为某类问题提供更高阶的精度和更高的效率。(2)三维正交各向异性孔隙弹性波传播问题的求解，并应用于体外冲击波治疗（ESWT）的持续研究。(3)发展了用于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)开发多层蒙特卡罗能力，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。这些问题是用偏微分方程来模拟的，而偏微分方程不能精确地解决，因此必须在数百万个网格点上计算生成数值近似。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。大部分工作都是非常通用的，可以应用于许多不同的问题，算法是在开源软件包Clawpack中实现的，该软件包可在www.clawpack.org上免费获得。这项拨款所支持的工作是由两个具体的实际应用所强烈推动的。一个是模拟危险的地球物理流动，特别是海啸，以及开发更迅速、更准确地预测海啸影响的技术。就风险评估和减轻灾害而言，还必须考虑未来可能引起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所造成的淹没概率。另一个激励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，这可能导致生物反应，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很重要的，这是一种临床治疗方法，已经显示出一些显著的结果。

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.