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

研究者继续发展自适应移动网格方法的数值解决时间依赖，多维偏微分方程。研究工作将集中在一种新的移动网格方法上，即研究者及其合作者提出的移动网格偏微分方程方法。该方法已在一维和二维上实现，用于生成非奇异的结构化和非结构化自适应网格，并成功地应用于许多问题。此外，该方法还形成了描述以前方法的统一框架，提供了新的理论基础，并建立了可靠的新方法。本文的目标是进一步提高二维方法的效率和鲁棒性，将其应用于实际问题，并实现三维方法。该项目关注的是新的计算方法的发展，这些方法对于提高科学家和工程师解决对我们的经济、环境和安全至关重要的大规模计算问题的能力至关重要。研究的重点是发展自适应数值技术或网格自适应方法，以适应所解决的特定问题的特殊运动特征。近年来，网格自适应在科学、工程和工业领域的许多大规模问题的数值模拟中发挥了不可或缺的作用，例如涉及激波、边界层、点火传播前沿和多材料界面的问题。这些问题有一个明显的共同特征，即它们的解仅在物理域的一小部分发生显著变化，并且该部分解的分辨率决定了整个模拟的质量。标准的（非自适应的）技术常常不能解决这些问题，因为它们平均地在整个域上花费精力，因此需要大量的计算机cpu时间和内存资源来获得合理程度的分辨率。另一方面，自适应网格方法通过最关注解决方案变化最大的物理域的一小部分而获得显著的经济性。所研究的移动网格方法是一种自然类型的自适应网格方法，旨在捕捉物理解的运动特征。该方法适用于并行计算，已被证明是许多工业制造问题仿真中不可或缺的工具。作为拟议研究项目的重要组成部分，将重点关注两个具体的应用。首先是地下水含水层中化学物质运移的数值模拟。在美国，地下水供应了大部分的用水。化学污染对地下水水质的广泛影响，促使人们对地下化学行为的模拟和预测进行了广泛的研究。移动网格方法的应用将为模拟地下水中化学物质的运移提供准确、高效和鲁棒的数值算法，从而有效地保护和管理地下水资源。另一个应用将是分析翼型的动态失速，以便更好地理解导致现代战斗机和民用运输机在大迎角飞行条件下非定常流动行为的物理机制。

The investigator continues to develop adaptive moving mesh methodsfor the numerical solution of time dependent, multi-dimensionalpartial differential equations. The research work will be focusedon a new moving mesh method, the moving mesh partial differentialequation approach proposed by the investigator and his collaborators.The approach has been implemented in one and two dimensions forgenerating non-singular structured and unstructured adaptive meshesand successfully applied to a number of problems. Moreover,the approach has led to a unifying framework describing previousmethods, providing a new theoretical underpinning, and buildingreliable new methods. The objectives of the proposal are tofurther improve the efficiency and robustness of the two dimensionalmethod, to apply it to practical problems, and to implementthe three dimensional method.This project is concerned with the development of new computationalmethods which are essential to enhance the ability of scientists andengineers to solve large scale computational problems that are crucialto our economy, environment, and security. The research is focused ondevelopment of adaptive numerical techniques or mesh adaptation methods,where the special moving features of the particular problem being solvedare adapted to. Mesh adaptation has recently played an indispensable rolein the numerical simulation of many large-scale problems arising fromscience, engineering, and industry, such as those involving shockwaves, boundary layers, ignition propagation fronts, and multi-materialinterface. These problems have a distinct common feature, that is, theirsolution changes significantly only in a small portion of the physicaldomain and the resolution of the solution in this portion dominatesthe quality of the whole simulation. Standard (non-adaptive) techniquesoften fail to solve these problems because they spend effort evenly onthe entire domain and thus require formidable resources of computerCPU time and memory to obtain a reasonable degree of resolution.On the other hand, adaptive mesh methods gain significant economiesby paying most attention to the small portion of the physical domainwhere the solution changes most. The moving mesh method under studyis a natural type of adaptive mesh methods which are designed tocapture the moving features of the physical solution. The methodis suitable for parallel computing and has proven to be an indispensabletool for use in the simulation of many industrial manufacturing problems.As an important part of the proposed research project, two specificapplications will be focused on. The first will be the numericalsimulation of chemical transport in groundwater aquifers.Groundwater supplies much of the water use in the UnitedStates. The wide spread degradation of groundwater quality fromchemical contamination has recently prompted extensive research forsimulating and predicting chemical behaviors in the subsurface.The application of the moving mesh methods will provide accurate,efficient, and robust numerical algorithms for simulating chemicaltransport in groundwater and therefore for effectively protectingand managing the groundwater resources. The other application will beon the analysis of dynamic stall of airfoil for better understandingthe physical mechanisms which cause the unsteady flowbehavior in the high-angle-of-attack flight condition found commonwith modern fighter and civil transport aircrafts.