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

该项目致力于设计和研究在理论和计算上有效的方法来数值求解对流起重要作用的问题。许多情况都属于这一类，虽然我们提议的发展可以很容易地适用于大多数情况，但我们将把我们的努力集中在其中两种情况上。第一个以浅水方程为模型，研究飓风预报和港口环境研究；第二个，以哈密尔顿-雅各比方程为模型，研究机器人车辆的地形导航(最短时间穿越路径的计算)和集成电路制造中的材料刻蚀。该项目的主要重点是设计所谓的后验误差估计，这是数学上合理的惠普自适应算法的基础。我们考虑了如何高效地获得对几个实际感兴趣的物理现象的高精度计算机模拟的问题，即墨西哥湾的飓风预报和港口的环境研究，以及集成电路制造中机器人车辆的最短时间穿越路径的计算和材料刻蚀。由于这些复杂问题的精确解尚不清楚，为了保证给定的模拟精度，必须设计适当的特殊技术来评估其质量。此外，这些技术可以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模拟；这样，其计算效率可以显著提高。

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.