Moving Mesh Methods for Numerical Solution of Time Dependent Partial Differential Equations in Two and Three Spatial Dimensions

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

The investigator continues to develop adaptive moving mesh methods for the numerical solution of time dependent, multi-dimensional partial differential equations. The research work will be focused on a new moving mesh method, the moving mesh partial differential equation approach proposed by the investigator and his collaborators. The approach has been implemented in one and two dimensions for generating non-singular structured and unstructured adaptive meshes and successfully applied to a number of problems. Moreover, the approach has led to a unifying framework describing previous methods, providing a new theoretical underpinning, and building reliable new methods. The objectives of the proposal are to further improve the efficiency and robustness of the two dimensional method, to apply it to practical problems, and to implement the three dimensional method. This project is concerned with the development of new computational methods which are essential to enhance the ability of scientists and engineers to solve large scale computational problems that are crucial to our economy, environment, and security. The research is focused on development of adaptive numerical techniques or mesh adaptation methods, where the special moving features of the particular problem being solved are adapted to. Mesh adaptation has recently played an indispensable role in the numerical simulation of many large-scale problems arising from science, engineering, and industry, such as those involving shock waves, boundary layers, ignition propagation fronts, and multi-material interface. These problems have a distinct common feature, that is, their solution changes significantly only in a small portion of the physical domain and the resolution of the solution in this portion dominates the quality of the whole simulation. Standard (non-adaptive) techniques often fail to solve these problems because they spend effort evenly on the entire domain and thus require formidable resources of computer CPU time and memory to obtain a reasonable degree of resolution. On the other hand, adaptive mesh methods gain significant economies by paying most attention to the small portion of the physical domain where the solution changes most. The moving mesh method under study is a natural type of adaptive mesh methods which are designed to capture the moving features of the physical solution. The method is suitable for parallel computing and has proven to be an indispensable tool for use in the simulation of many industrial manufacturing problems. As an important part of the proposed research project, two specific applications will be focused on. The first will be the numerical simulation of chemical transport in groundwater aquifers. Groundwater supplies much of the water use in the United States. The wide spread degradation of groundwater quality from chemical contamination has recently prompted extensive research for simulating and predicting chemical behaviors in the subsurface. The application of the moving mesh methods will provide accurate, efficient, and robust numerical algorithms for simulating chemical transport in groundwater and therefore for effectively protecting and managing the groundwater resources. The other application will be on the analysis of dynamic stall of airfoil for better understanding the physical mechanisms which cause the unsteady flow behavior in the high-angle-of-attack flight condition found common with modern fighter and civil transport aircrafts.