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

本项目致力于设计和研究在理论上和计算上有效的方法来解决对流扮演重要角色的数值问题。很多情况都属于这一类，尽管我们提出的开发方法可以很容易地应用于大多数情况，但我们将把精力集中在其中两种情况上。第一个以浅水方程为模型，是对飓风预报和港口环境研究的研究；第二个以汉密尔顿-雅可比方程为模型，是对机器人车辆的地形导航（计算最小时间传输路径）和集成电路制造中的材料蚀刻的研究。该项目的主要焦点是设计所谓的后验误差估计，这是数学上合理的hp自适应算法的基础。我们考虑的问题是如何有效地获得几个实际感兴趣的物理现象的高精度计算机模拟，即墨西哥湾的飓风预报和港口的环境研究，以及机器人车辆的最小时间传输路径的计算和集成电路制造中的材料蚀刻。由于这些复杂问题的确切解是未知的，为了保证模拟的给定精度，必须适当地设计特殊技术以评估其质量。此外，这些技术可用于自动让计算机知道何时何地增加或减少计算工作量以获得仿真；这样可以显著提高其计算效率。

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.