Course Description:

Many application problems leads to partial differential equations (PDE) in which analytic solutions are rarely available or too complicated. The rapid development of modern computers has made it possible to solve many important application problems numerically. Numerically simulations, on the other hand, provide useful, sometimes vital information, to better understand the original problem in terms of modeling, optimization, and theoretical study.

This course is designed to cover materials that are not covered in other numerical analysis courses. Particularly, the applications include problems in fluid mechanics, electro-magnetics, mathematics biology, image processing, and inverse problems. These problems may involve irregular domains, phase
changes, discontinuities in physical parameters, free boundary and moving interface problems. We will discuss how to use simple finite difference/finite element methods based on simple grid structure, like Cartesian grids to solve the application problems efficiently.

Some state of the art techniques such as harmonic averaging, smoothing method, Peskin's immersed boundary method, the immersed interface method, and the level set methods will be covered in this course. Most of these methods are simple to implement but powerful to solve some important problems. We will explain these methods with applications.
 

A reasonable background in linear algebra, numerical analysis, partial differential equations, and finite difference methods.

• The Immersed Interface Method -- Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, Zhilin Li and Kazufumi Ito, SIAM Frontiers in Applied mathematics, 33 (optional)

Notes from the instructor, and references from the literature. Below are some important ones:

Level Set Methods and Dynamic Implicit Surfaces, S. Osher and R. Fedkiw, Springer Press, 2002.

Level Set Methods and Fast Marching methods, J. Sethian, Cambridge University Press, 2nd Ed., 1999

The immersed boundary method, C. S. Peskin, Acta Numerica, 1-39, 2002.

Beyer_LeVeque

Peskin_Review

WENO and Level Set Method (G-S Jiang & D. Peng, SIAM J. Sci. Comput., Vol. 21, No. 6, pp. 2126-2143, 2000)

Grading:

There may be some suggested homework assignments or computer labs. Each participant select a project of his/her own interest (can from the literature as well) and present it to the entire class at or near the end of the semester. Class participation is strongly encouraged.

Course web-page:

 http://www4.ncsu.edu/~zhilin/TEACHING/MA798/

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