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

获奖摘要：奖项编号：0209313PI：曹维明机构：德克萨斯大学圣安东尼奥分校项目：计算数学项目经理：Catherine mavriplis标题：平流扩散方程数值解的自适应网格细化偏微分方程数值解的一个基本问题是自适应网格细化。局部细化（h方法）和移动网格方法（r方法）是两种基本的方法。每一种都有其特点。h-方法具有鲁棒性和可靠性。对于时间相关的问题，r-方法是非常有效的。该项目的目标是开发一种综合的h-和r-细化方法，以实现两种方法的优势。这种组合方法特别适用于求解平流扩散问题，可以显著降低数值扩散和平流速度。本研究需要解决的主要问题包括：指导hr精化的误差估计、与空间hr精化相结合的稳定精确的时间积分方法；基于精细网格的保守数值格式；离散化引起的线性代数方程的有效解实现和软件开发。虽然分析和算法将在一般时间依赖的平流扩散问题的背景下建立，但我们将重点关注两个具体应用：环境科学中地下水系统的污染和修复，以及生物物理学中白细胞的运动和变形。随着现代计算机技术的进步，数值实验现已成为科学、工业和工程领域的两大主要工具之一（另一个是物理实验）。许多重要过程的数值模拟都涉及到平流扩散问题的求解。例如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修复以及生物医学设备的创新。本项目旨在开发一种鲁棒、可靠、高效的网格细化方法，用于求解平流扩散问题。本文将讨论该方法的设计、分析和实现中的主要问题。特别是，它将直接应用于解决环境和生物物理问题。我们预计，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，而且将产生高效、精确的数值算法，这些算法将应用于对我国国民经济和安全至关重要的各种工业和工程领域。日期：2002年5月22日

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