Math 615 Numerical Analysis of Differential Equations
Spring 2014 UAF
Course Description: 3.0 credits.
We cover methods for numerical approximations to partial differential
equations (PDEs) and related problems on computers. There will be some coverage of
numerical methods for ordinary differential equations (ODEs). PDEs are
the underlying structure for most problems of flow, fields, thermodynamics,
deformation, quantum mechanics, chemistry, and so on, but also finance and population dynamics.
We do mathematical analysis of these numerical methods, with both
practical and abstract approaches to this numerical analysis: (i) how
are these methods implemented, verified, and used? (ii) when are they
stable? and (iii) do we know in advance that they converge?
The emphasis will be on finite difference methods for PDEs. I will only
gloss other methods (e.g. finite volume and element) for PDEs.
Lectures will include Matlab
demonstrations whenever I can fit them in. I will help you get started
with Matlab, but you must show
initiative in learning actual numerical computation.
Homework assignments and a student-chosen project will include actual
implementations in Matlab.
Abstract and precise thought is, however, essential to make choices
among numerical methods for solving major problems. Thus all homework
assignments will have mathematical exercises, and in these you will be asked
to "show" and "prove". Formal proof style is not important, but you'll
need to give clear presentations of sufficiently-general logical arguments.
We will think in terms of vectors and
matrices. We will not be satisfied with seeing lots of numbers or pretty
pictures from our computer programs. Instead we will be concerned with
the degree to which our numbers represent what we claim they approximate. We
will seek the underlying linear algebra structure of PDE problems. The
course will include some nonlinear examples, for which one uses a
sequence of approximating linear problems.
Goals: At the end this course you will
not be a professional numerical analyst. But you will be able to
evaluate and use many numerical tools for solving scientific and engineering
problems, and you will be able to code some of the basic methods (i.e. for
the purpose of prototyping more serious solutions). Furthermore you
will have the mathematical start needed to take the next steps to learn the
finite element method, spectral methods, matrix methods, and broader
Calendar and Course Webpage:
A day-to-day (tentative) schedule for the semester is at the course webpage bueler.github.io/M615S14/.
undergraduate ordinary differential equations, undergraduate linear
algebra, exposure to the basic ideas of numerical analysis, and exposure to
Fourier series and separation of variables (for solving the classical linear
PDE boundary value problems). Also some exposure to computer
programming. Formally: "Prerequisites: CS F201, MATH F310, MATH F314, MATH F421, MATH F422 or permission of instructor."
Textbook: The required
text is K. Morton and D. Mayers, Numerical
Partial Differential Equations, 2nd ed. Cambridge Univ. Press 2005.
We will cover
chapters 1, 2, 3, 4, 5, and 6.
Your Grade = Homework + Project
+ Midterm: Sixty
percent of the course, and the grade, will be based on weekly
homework assignments, including a final assignment which will be
worth double an ordinary assignment. You will be asked to think abstractly on
some problems and to use Matlab on others. (Expect to write programs
about a half-page long.)
It is assumed that students in this class have in mind,
or can acquire, specific modelling problems in applied fields
which can be used for a project.
These will mostly, but not exclusively, be PDE problems. Students
often use a simplification of a thesis/dissertation project, for
instance. I am eager to help and advise on choosing and refining such
percent of the grade in the course will be on the project. Two
project assignments will be given, the first part a preparatory stage due
midsemester, and the remainder due at finals time. Both mathematical
analysis and actual numerical computation are required on your project.
Finally, there will be a one-hour in-class midterm
exam worth only fifteen percent
of the course grade. The purpose is to give a midsemester opportunity
to review basics before expanding our goals in the second half.
|The course grade will be determined from
according to the schedule at right
I will use plus/minus grades as indicated.
91 --100 %
88 -- 90 %
84 -- 87 %
76 -- 83 %
73 -- 75 %
69 -- 72 %
57 -- 68 %
41 -- 56 %
0 -- 40 %
Policies and makeup exams: The department has
specific policies on incompletes, late withdrawals, and early final
examinations, etc; see http://www.uaf.edu/dms/policies/
by the UAF Honor Code. I will work with the Office of Disabilities
Services (208 Whitaker, 474-5655) to provide reasonable accommodation to
student with disabilities.
Programming in the course: You
will use Matlab, Octave, or pylab
(= python+scipy+matplotlib) in this course. These will be
used in homework problems and in projects. Matlab is commercial while Octave
and Pylab are free and open
source. Octave is a clone of
Matlab so the same programs will
run in both. Programs in Matlab/Octave will appear on my website (and
occasionally in pylab).
Copious online resources are available
for learning Matlab/Octave/pylab. They are all languages designed to do numerical analysis
coursework. Mathematical and graphical inputs and outputs are easily
manipulated. Matrices appearing in problems can be easily
analyzed. Many of the operations appearing in numerical problems are
natural and quick and require less work than in compiled
languages like C or FORTRAN.