Adaptive hr-mesh refinement for the numerical solution of advection-diffusion equations

平流扩散方程数值解的自适应 HR 网格细化

• 批准号: 0209313
• 负责人: Weiming Cao
• 金额: $ 9.03万
• 依托单位: University of Texas at San Antonio
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0209313&HistoricalAwards=false
• 关键词: Adaptive; hr; mesh; refinement; numerical

DMS Award AbstractAward #: 0209313PI: Weiming CaoInstitution: University of Texas, San AntonioProgram: Computational MathematicsProgram Manager: Catherine MavriplisTitle: Adaptive hr-mesh refinement for the numerical solution of advection-diffusion equationsA fundamental issue in the numerical solutions of partial differential equations (PDEs) is adaptive mesh refinement. Local refinements (the h-methods) and moving mesh methods (the r-methods) are two basic types of approaches. Each has its features. The h-methods are robust and reliable. The r-methods can be highly efficient and effective for time dependent problems. The goal of this project is to develop a comprehensive combined h- and r-refinement method to achieve the advantages of both approaches. This combined method is especially useful in solving advection-diffusion problems, with which the numerical diffusion and advection speed can be significantly reduced. Major issues to be addressed in this research include: the error estimates used to guide the hr-refinement, stable and accurate time integration methods used in conjunction with the spatial hr-refinement; the conservative numerical schemes based on the hr-refined meshes; the efficient solution of the linear algebraic equations arising from the discretization; the implementation and software development. While the analysis and algorithms will be established in the context of general time dependent advection-diffusion problems, we shall focus on two specific applications: the contamination and remediation of ground water systems in environmental science, and the motion and deformation of white blood cells in biophysics.With the advances of modern computer technology, numerical experiments have now become one of the two primary tools in science, industry, and engineering (the other is physical experiments). Numerical simulation of many important processes involves the solution of advection-diffusion problems. Examples include optimal design of aircrafts and automobiles, forecast of global weather change, remediation of surface and ground water contamination, and innovation of biomedical devices. This project aims to develop a robust, reliable, and highly efficient mesh refinement method used for solving the advection-diffusion problems. Major issuesin the design, analysis, and implementation of this method will be addressed. In particular, it will be applied directly to solve environmental and biophysical problems. We anticipate that successful completion of this project will not only enrich the mathematical theory of adaptive solution of partial differential equations, but also generate highly efficient and accurate numerical algorithms, which will find applications in a variety of industrial and engineering areas that are essential to our national economy and security.Date: May 22nd, 2002

DMS Award AbstractAward #： 0209313 PI： 曹伟明研究机构： 德克萨斯大学圣安东尼奥分校 计算数学项目经理：Catherine Mavriplis题目：对流扩散方程数值解的自适应hr网格加密偏微分方程（PDE）数值解的一个基本问题是自适应网格加密。局部加密（h-方法）和移动网格方法（r-方法）是两种基本类型的方法。 各有各的特点。h-方法是鲁棒的和可靠的。r-方法对于时间相关问题可以是非常高效和有效的。这个项目的目标是开发一个综合的组合h-和r-细化方法，以实现这两种方法的优点。这种组合方法在求解对流扩散问题时特别有用，数值扩散和对流速度可以大大降低。本研究所要解决的主要问题包括：用于指导hr-细化的误差估计，与空间hr-细化结合使用的稳定和精确的时间积分方法;基于hr-细化网格的保守数值格式;离散化所产生的线性代数方程组的有效解;实施和软件开发。虽然分析和算法将建立在一般的时间依赖对流扩散问题的背景下，我们将集中在两个具体的应用：环境科学中的地下水系统的污染与修复，生物物理学中的白色血细胞的运动与变形，随着现代计算机技术的发展，数值实验已成为科学、工业、和工程（另一个是物理实验）。许多重要过程的数值模拟都涉及对流扩散问题的求解。例子包括飞机和汽车的优化设计，全球天气变化的预测，地表水和地下水污染的补救，以及生物医学设备的创新。本计画旨在发展一个强健、可靠且高效率的网格加密方法，以解决对流扩散问题。主要问题在设计，分析和实施这一方法将得到解决。特别是，它将直接用于解决环境和生物物理问题。我们期望，该项目的成功完成不仅将丰富偏微分方程自适应解的数学理论，而且还将产生高效和精确的数值算法，这些算法将在对我们的国家经济和安全至关重要的各种工业和工程领域中得到应用。