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 interdisciplinary collaborations and will be integrated with education at 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.
