Original announcement of the project presentations: This class included a very substantive project that each student was supposed to work on for at least about six weeks. Over two days of classes (on May 08 and 10), the projects will be presented; see the program for project presentations for details. During that time, we will also perform a peer-review (among the students in the class) of the reports. The final, edited reports will be posted as postscript files and linked through the program for project presentations.
However, despite their many forms, many equations share certain fundamental mathematical properties and can be classified into the three basic categories of elliptic, parabolic, and hyperbolic partial differential equations. It makes therefore sense to study the mathematical properties and numerical methods for prototype equations of each type.
This class will provide an overview of the types of equations, their most fundamental mathematical properties, and demonstrate numerical methods for them. The most basic methods are finite difference and finite element methods, and we will discuss both types for each prototype equation. We will use this as the basis for discussing the associated issues of discretizing the time-direction and solving large sparse systems of linear equations efficiently with respect to memory and computing time.
It is not possible to cover this subject in one semester. Hence, students should be prepared to do some independent research as part of the homework. The class will also entail a very substantive final project involving a much broader problem than a homework would, for instance, an engineering design problem, where the numerics are used to answer a question quantitatively. The final project will culminate in a written report and an oral presentation on the level of a capstone experience for a Master of Science degree without a thesis.
To allow sufficient time for work on the final project and to enable the attendance of part-time students, the lectures will be mostly concentrated in the first half of the semester, while lengthened to two hours; see the syllabus for details.