A-Posteriori-Error-Estimates-Based Numerical Methods for Shallow Water and Hamilton-Jacobi Equations

基于后验误差估计的浅水和 Hamilton-Jacobi 方程的数值方法

• 批准号: 0107609
• 负责人: Bernardo Cockburn
• 金额: $ 12万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0107609&HistoricalAwards=false
• 关键词: Posteriori; Error; Estimates; Based; Numerical

This project is devoted to devising and studying, theoretically as well as computationally, efficient methods for numerically solving problems in which convection plays a significant role. A wide range of situations falls into this category and, although what we propose to develop can be easily applied to most of them, we are going to focus our efforts into two of them. The first, modeled by the shallow water equations, is the study of hurricane forecasting and of environmental studies in ports, and the second, modeled by the Hamilton-Jacobi equations, is the study of terrain navigation (computation of minimum time transit paths) of robotic vehicles and material etching in integrated circuit fabrication. The main focus of the project is the devising of the so-called a posteriori error estimates that are the basis for mathematically sound hp-adaptive algorithms.We consider the problem of how to efficiently obtain highly accurate computer simulations of several physical phenomena of practical interest, namely, hurricane forecasting in the Gulf of Mexico and environmental studies in ports, and the computation of minimum time transit paths of robotic vehicles and material etching in integrated circuit fabrication. Since the exact solution of these complex problems is not known, in order to guarantee a given accuracy of the simulation, special techniques have to be suitably devised in order to assess its quality. Moreover, these techniques can be employed to automatically let the computer know when and where to increase or decrease the computational effort to obtain the simulation; in this way, the efficiency of its computation can be significantly enhanced.

该项目致力于设计和研究，理论上以及计算上，有效的方法，数值求解问题，其中对流起着重要的作用。许多情况福尔斯都属于这一类别，虽然我们建议制定的办法可以很容易地适用于其中大多数情况，但我们将集中精力处理其中两种情况。第一，浅水方程建模，是飓风预报和港口环境研究的研究，第二，由哈密尔顿-雅可比方程建模，是地形导航（计算最小时间过境路径）的机器人车辆和集成电路制造中的材料蚀刻的研究。该项目的主要重点是设计所谓的后验误差估计，这是数学上健全的hp自适应算法的基础。我们考虑的问题是如何有效地获得高度准确的计算机模拟的几个物理现象的实际利益，即，飓风预报在墨西哥湾和港口环境研究，以及集成电路制造中机器人车辆和材料蚀刻的最小时间通过路径的计算。由于这些复杂问题的确切解决方案是未知的，为了保证给定的模拟精度，必须适当地设计特殊的技术，以评估其质量。此外，这些技术可以用来自动让计算机知道何时以及在何处增加或减少计算工作量以获得模拟;以这种方式，其计算效率可以显着提高。