[ Home | Contact Information | Office Hours | Weekly Schedule | Class Web Pages | Curriculum Vitae | Conference Participation | Research Information | Publications | Past Ph.D. Students | Mathematical Lineage | History of Kinetic Theory | Potpourri ]

Mapl 612 Description and Prerequisites

UM Graduate Catalog Course Description

Mapl 612, Numerical Methods in Partial Differential Equations (3 credits) Finite difference methods for elliptic, parabolic, and hyperbolic partial differential equations. Additional topics such as spectral methods, variational methods for elliptic problems, stability theory for hyperbolic initial-boundary value problems, and solution methods for conservation laws.

Course Prerequisites

More Details

This course is intended for both students in mathematics and students in applied fields. It covers the mathematical foundations of many numerical methods used by computer codes for simulations in aerospace, astrophysics, semiconductor design, and other fields of application. The material should be useful to students who use such codes, develop their own codes or variants, or who wish to work on the mathematical theory.

There will be no exams. There will be four homework assignments each of which will have a theoretical part and a computational part. The computational parts may be completed in a reasonable (Matlab, Fortran, C, C++, etc.) programming language of your choice. Your course grade is determined primarily by your performance on the homework.

The lectures will be partly theoretical and partly practical. They will cover the basic material in the field pertaining to linear elliptic, parabolic, and hyperbolic partial differential equations. They will also cover numerical methods for nonlinear equations that have certain natural mathematical or physical structures such as variational problems, conservation laws, and Hamiltonian dynamics. The relationship between consistency, accuracy, convergence, and (especially) stability will be emphasized. They will also emphasize how the mathematical or physical structure of a problem should influence how to devise numerical methods. With regard to the various linear algebra strategies that are used to implement some of these methods, theory will be treated lightly while practical considerations will be emphasized. Time permitting, the basics of multigrid methods may be covered.

Some knowledge of partial differential equations (PDE), either directly or through courses such as fluid mechanics or semiconductor design that use PDE extensively. Programming experience of some kind is almost a prerequisite, although it is possible to learn as we go along if one makes an extra time committment. Also useful, but not absolutely necessary, is some experience with elementary numerical analysis at the undergraduate level (e.g., MAPL 460 or 466 or equivalent). Please feel free to contact me concerning your background if you have any questions.