Math 615 (Applied) Continuum Numerical Analysis

Ed Bueler, Spring 2007 UAF

Instructor:     Ed Bueler       Chapman 301C
Phone: 474-7693   eMail: ffelb@uaf.edu
Office Hours:    MW 2:15-2:15, F 9:15-10:15
 Class Time: MWF 1:00--2:00
Classroom: Chapman 107.
Web Site: http://www.dms.uaf.edu/~bueler/

Course Description:  TOPICS: Methods for approximating partial differential equations (PDEs) and related problems on computers.  Mathematical analysis of these methods.  Abstract frameworks for understanding numerical analysis of continuum problems.
     CHAPTERS:  1, most of 2, most of 3, most of 4, first half of 5, some of 6, little of 7.
     Most classtime will be spent with my lecture, with Matlab demonstrations when I can fit them in.  I will schedule a couple of days in the Chapman 103 lab to get you started with Matlab, though you will be mostly on your own there.
     You will do both practical and abstract approaches to problems.  Homework assignments and a student-chosen project will be practical and will require actual implementation (in Matlab).  Abstraction is essential in order to understand the choices one faces, and all homework assignments will have mathematical exercises.
      The emphasis is on finite difference methods, but I will also gloss spectral methods and finite elements.  Frequent emphasis on thinking in matrices.  Instead of a list of finite difference schemes, for instance, the theme might be how to see the underlying matrix structure.  The course will include exposure to real nonlinear examples.  (One always replaces such problems with a sequence of approximating linear problems.)

Prerequisites:  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.  Matlab experience is desirable but not essential if you are a fast learner and have done other programming.
     Formally, the prerequisites are MATH 302, MATH 310, MATH 314, and MATH 421 or permission of the instructor.  The list in the catalogue is in error.  CS 201 and MATH 422 are not specifically needed, though they are nice things to know.

Textbook:  The required text is

There are, of course, many other textbooks on numerical analysis of PDEs, but I actually like this one.  Four other texts are recommended, of which two are freely available a page at a time:

Grade = Project + Homeworks :  It is assumed that students in this class have in mind or can acquire specific continuum  modelling problems in applied fields.   These will mostly, but not exclusively, be PDE problems, and they are supposed to be nontrivial problems.  Frequently they are a component of (or a simplification of) a thesis/dissertation project.  I am eager to help and advise on choosing and refining such problems. Forty percent of the grade in the course will be on a project based on such a problem. Two project assignments will be given, one due midsemester, and one due at the finals time.  In both cases, actual numerical computation will be required, generally in Matlab.  It is expected that the first part will be preparatory for the more complete second part.  Furthermore, at least one presentation of the project will be required during the semester.  The presentation can be either oral or on a poster.  These presentations are important to the class, as the class will act as consultants to the presenter.
     The other sixty percent of the course (and grade) will be based on (nearly) weekly homework assignments.  Here is where students will be encouraged to understand the mathematics and gain breadth and perspective.  Students will be encouraged to take a matrix/vector view of the structure of these problems.  You will use Matlab on many of these exercises, too.   You will be expected to crank out at most a couple of half-page-long Matlab programs per homework assignment.  The last homework assignment will be worth double the previous assignments; it may be regarded as a take-home final exam.

Policies and makeup exams:   The department has specific policies on incompletes, late withdrawals, and early final examinations, etc; see http://www.dms.uaf.edu/dms/Policies.html .  You are covered by the UAF Honor Code.  I will work with the Office of Disabilities Services (203 WHIT, 474-7043) to provide reasonable accommodation to student with disabilities.

Programming in the course:  We will use Matlab.  UAF has a campus-wide site license so you should be able to use it for free.  It is available in the Chapman 103 and other computer labs on campus.  Programs in Matlab will appear on my website, and these can be used in homework problems and in projects. Copious resources are available for learning Matlab and programming therein, including my modest tutorial and links page (www.math.uaf.edu/~bueler/MatlabEx.htm), but students with no programming experience will have a high hurdle to cross.  The programming experienced in Math 310 should be sufficient as preparation, however.  Students who are very well-established and secure in some other language, e.g. Python, may use it.  Use of other languages is fully the students responsibility, and, in fact, may cause substantial disadvantage.  

An ad for Matlab might look like this:  Matlab is a language designed to do numerical analysis coursework!  Programs can be written and run in Matlab in a highly traditional programming style.  Mathematical and graphical inputs and outputs can be handled more directly in Matlab than in most compiled programming languages.  Matrices appearing in problems can be easily analyzed.  Many of the operations appearing in numerical problems are natural and quick in Matlab, and require much more work in C or FORTRAN.  Even established compiled-language programmers will find it a desirable prototyping tool.