Your Project

Math 615 Applied Numerical Analysis, Ed Bueler, Spring 2007 UAF

Overview:  My goal is for you to get practical experience in numerical analysis of continuum problems by working on a project of direct interest to you.  As you are all graduate students, the obvious advice is to pick a project in consultation with your advisor; see the "subject" comment below.  You will submit a one page Proposal on Monday 2/26; a short Version 1.0 on Friday 3/30; and a final  Version 2.0 on Friday 5/11; see comments on length and format below.

Subject of Your Project:  You should pick a continuum mathematical model like

You must identify which one of these it is!  Most ordinary differential equation problems lead to numerical analysis rather unlike the content of the course, and thus are not recommended. 

You will do a numerical analysis of a particular problem for your model.  The problem could be a

You must identify, early in your project, which one of these it is!


Content of Your Project:  You will do a numerical analysis, but with a computed example as well, for your chosen scheme or schemes applied to your continuum mathematical model. In particular, three things are required:
  1. numerical analysis
  2. practical computation
  3. discussion of scientific/engineering meaning (including conclusions about the role/meaning of your particular computation)
Note the first part: You must make a serious attempt to understand and explain the efficiency, stability, and accuracy of your numerical scheme(s).  In particular, if it is possible and meaningful to do so you should
On the other hand, I do not expect you to derive the equations of the continuum model (typically the PDEs).  That is the subject for other courses.  Instead, careful statement of the equations of the model, explanation of the symbols, and careful references suffice.  You should state the type of the problem (i.e. "..an initial/boundary problem for a nonlinear evolution equation of mixed parabolic and elliptic type..." or "... an inverse problem for a linear, elliptic boundary value problem..." or "... a nonlinear stationary variational inequality ...") as precisely as possible.

I strongly recommend that you do the practical computation in Matlab, and I do not expect you to produce a production quality code.  Rather, your goal should be a functional and readable prototype.  (Something you can build upon!)  Please use Matlab's built-in linear algebra, and don't write your own!  You don't need to use Matlab tricks, but you may find that learning to vectorize and avoid loops will help you understand.

You are encouraged to look for methods other than finite differences for your practical computation.  In particular, consider
methods.  These methods differ both in implementation and in their analysis.

In what follows I state what you will turn in at various stages, but some of the work is mine, too.  I will be giving you feedback on each part.  You can and should also come to me and talk about your project at any stage.


Length and Format of Proposal (due Monday 2/26/07):  The proposal should be one page and should include at least three references.  (If you find it hard to pack it on to one page, then think of this as practice for extracting money from funding sources; their length requirements can be much more painful.)  Briefly state the continuum model, the scientific/engineering context, the type of the problem (see above), and the numerical method you propose to apply.  Then state what forms of analysis of the method you think are feasible; you need not include that analysis yet.  At least one of your references should address the numerical analysis of your chosen (or closely related) model, problem, and method.


Length and Format of Project Version 1.0 (due Friday 3/30/07):  The point here is to get you to do a little bit of everything that will go into the final product.  Version 1.0 should be a very good skeleton on which to build Version 2.0.  It needs to be readable but it need not be polished.

The length should be between 5 and 15 pages total.  Please include the following 7 section headings (meaning in parentheses):

  1. Introduction (including scientific/engineering context and sketch of the numerical analysis you do, especially the high points)
  2. Continuum Model and Problem (here should go the PDE, etc, and clear specification of the particular problem including boundary conditions and parameters; include one or two exact solutions if known)
  3. Numerical Scheme (state the method, discuss the reasons for this choice, discuss implementability and efficiency; excerpts of the code may go here)
  4. Analysis (truncation error, stability, convergence, cost of algorithm)
  5. Results or Conclusion
  6. References
  7. Appendix/Appendices (put the actual codes here, tangential analysis or computation, failed attempts if significant)
The Results/Conclusion part should be rather short in version 1.0 because you are likely not able to draw definitive conclusions yet.


Length and Format of Project Version 2.0 (due Friday 5/11/07):  Now the point is to fill out the skeleton, of course.  You will undoubtedly need to alter your earlier plans, so don't worry if version 2.0 has different computational or analytical details than version 1.0.  Version 2.0 is the final version, and I encourage you to make it a little bit clean-and-neat.  Please use ordinary 8.5 by 11 paper and staple it in the upper left corner.  Please do not put it in a folder, etc.  (I will have a stack of these things to grade and I don't need yours to take up extra space. The length should be between 15 and 30 pages total and should include exactly the same sections as described for Version 1.0.

2/14/07 ELB