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

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

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.