# Mapl 612 Description and Prerequisites

** 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 *

- either a graduate level one semester course in partial
differential equations, a theoretical graduate level
course in an applied field such as fluid mechanics, or
permission of instructor.

## * 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.