Course Description:   Methods for approximating partial differential equations (PDEs) and related problems on computers.  How to do it in practice and how to think about the easy problems more abstractly as a way to understand the choices one faces in hard problems.

Emphasis on projects.  I want students to come with a serious, particular model in mind.  Lots of computed examples.  Students will use Matlab to build prototypes of numerical algorithms on many concrete examples.  Emphasis on thinking in matrices.  Instead of a list of finite difference schemes or a list of tedious programs, the theme will always be how to see the underlying matrix structure.  Exposure to nonlinear examples.  Real problems are nonlinear, but the scientific world functions by replacing such problems with a sequence of approximating linear problems.

Topics include:

• Introduction to some standard problems.
• The potential equation--which leads to:
• boundary value problems
• Fourier series and other spectral approximations
• finite differences
• eigenvalues of matrices
• sparse matrices
• Next, try out the heat equation, and study:
• finite differences: explicit and implicit
• method of lines
• stability
• Transport equations (like waves):
• flux-conservation
• upwind finite differences and stability
• Lax, Leapfrog, and Lax-Wendroff
• Some real-world and nonlinear examples:
• nonlinear transport
• ice flow
• Newton's method
• Schrödinger equation
• Some other ideas:
• finite elements
• inverse problems