This course is an introduction to numerical analysis and methods for the solution of partial differential equations. It covers modeling, fundamental discretization theory, error estimation, stability analysis, and approaches for the validation of the numerical codes. This course is practical in nature: the students will learn some theory, but mostly will engage in the solution numerous problems, including heat transport, fluid flow, differential geometry and image processing.
Most of the material from the course can be found in the following two books:
a) Numerical Methods for Partial Differential Equations, by W. Ames, third edition, Academic Press (1992).
b) Numerical solution of partial differential equations, by K.W. Morton and D. F. Mayers, Cambridge Univ. Press (2002).
c) Numerical Recipes: The Art of Scientific Computing, by William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling, Cambridge, (1997)
(This is the auxiliary text that you will need to complete the homework, you can read some of the chapters of this last book online at www.nr.com).
The grade will be based on two midterms, a final and special projects.
We will use Matlab. Little or no previous programming experience needed. The instructor will provide sample programs to "jump start" the learning process.
The complexity of the problems we will study resides on the mathematics, not on the programming.
Example of a project that we will do in this course: finite differences simulation of fluid flow in a box. (Click in the box with right button to add ink, and move mouse and left click to add momentum). Other projects involve image processing, modeling of shock waves, optimization problems, etc. (see details below).