Efficient Numerical Methods For Material Transport On Moving Interfaces And Hamilton Jacobi Equations

这个项目的目标是为两类重要的问题开发有效的计算算法。首先是发展具有全局动力学的运动界面上物质输运的有效数值方法。主要难点在于整体动力学、运动界面和界面上物质分布的耦合。研究人员将开发有效且稳健的方法，可以(1)跟踪移动界面上的物质传输，(2)将界面动力学与全局动力学相结合。特别地，所开发的数值方法将用于研究表面活性剂在两相流中的作用。二是分析和推广了快速扫描法这一近年来发展起来的求解矩形网格上Eikonal方程的有效迭代方法，并将其应用于非结构网格和一般的Hamilton-Jacobi方程。将进行收敛和误差分析。求解非线性问题的快速扫描方法在有限次迭代中收敛是一个显著的结果。在迭代方法的一般框架下对该方法的进一步探索，不仅将为许多重要的应用提供有效的数值方法，而且将为构建其他非线性问题的迭代方法提供见解。上述研究项目将涉及跨学科合作，并将与不同层次的教育相结合。数值计算在现代科学技术中起着至关重要的作用，而开发高效、鲁棒的数值算法是其潜在的基本任务。该项目旨在开发和分析两类具有挑战性的问题的有效数值算法，这些问题在流体，材料，生物学以及计算机视觉，最优控制和地球物理学中具有重要应用。

The objective of this proposed project is to develop efficient computational algorithms for two important classes of problems. The first is to develop efficient numerical methods for material transport on moving interfaces with global dynamics. The main difficulty is the coupling of the global dynamics, the moving interface and the material distribution on the interface. The investigator will develop efficient and robust methods that can(1) track material transport on moving interfaces,(2) couple interfacial dynamics with global dynamics.In particular the developed numerical methods will be used to study the effect of surfactants in two phase flow. The second is to analyze and extend the fast sweeping method, which is an efficient iterative method recently developed for Eikonal equations on rectangular grids, to unstructured grids and general Hamilton-Jacobi equations. Convergence and error analysis will be carried out. The fact that the fast sweeping method for a nonlinear problem converges in a finite number of iterations is a remarkable result. Further exploration of this method in the general framework of iterative methods will not only provide efficient numerical methods for may important applications but will also shed insight for constructing iterative methods for other nonlinear problems. The above research projects will involve interdisciplinarycollaborations and will be integrated with educationat different levels.Numerical computations play a crucial role in modern science and technology while development of efficient and robust numerical algorithms is the underlying basic task. This project is aimed to the development and analysis of efficient numerical algorithms for two classes of challenging problems with important applications in fluids, materials, biology as well as computer vision, optimal control, and geophysics.