|
|
|
Clear Excellence |
H |
87-100 points |
|
Entirely Satisfactory |
P |
70-87 points |
|
Low Passing |
L |
50-70 points |
|
Failed |
F |
0-49 points |
During the course, 7 homework assignments shall be given. Each homework shall contain 5 required topics and a bonus topic, each worth 1 point for a maximum of 6 points. The bonus topics are meant to allow for makeups of previous deductions. Homework assignments are given every second Tuesday and due two weeks thereafter. The purpose of the homework assignments is to achieve familiarity with the course material.
Four reading assignments shall be given, once every 3 weeks in a staggered order among the students. Assignments are given to a small group of students attending the course, typically in pairs. Each assignment is graded with 5 points. The purpose of the reading assignments is to introduce students to the tasks involved in research work. Points are awarded to this end as follows:
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1 point for reading the assigned text;
-
1 point for background research, i.e. placing the assigned text in the context of previous work;
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1 point for forward research, i.e. investigating the ramifications of the assigned text;
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1 point for working through some of the results from the assigned text;
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1 point for quality of presentation.
Reading assignments are presented to the class in the last 8 minutes of Thursday's class, starting on September 5. The purpose of the presentations is to instill the discipline of concise, informative talking in front of groups of colleagues. Of the 8 minutes, 5 minutes are allocated to presentation and 3 to questions and answers.
Revised grading policy
Reading assignments were an interesting experiment. However to do them right, much more time should be allocated and this would have been detrimental to coverage of a wide range of topics. The students having already presented reading assignments will receive the point credits as bonus points (2 points).
Also, homework assignments proved to be more time-consuming than first thought so only 5 assignments were completed.
The revised grading policy is:
(1) 50 points from homework (each homework is worth 10 points instead of the 5 originally intended)
(2) 30 points from final examination (final exam will have bonus questions for an extra 30 points)
(3) 20 points from project work
Points are translated to grades following the table above.
Course Texts
One of the tenets of graduate study is the ability to critically examine multiple sources in order to verify theories and to enhance understanding. Lecture notes for the course shall be provided online in Postscript and PDF formats. These are meant to document the progress of material presented in class. It is expected and required that the student supplement the lecture notes with reading from other sources.
Lecture notes
|
Lecture |
Topic |
Notes |
Lecture |
Topic |
Notes |
|
1, 8/20 |
Overview of PDE's |
10/10 |
Lab: advection |
||
|
2, 8/22 |
ODE/PDE problems |
15, 10/15 |
Modified equations |
||
|
3, 8/27 |
Weighted residual |
10/17 |
(Fall break - no class) |
||
|
4, 8/29 |
ODE IVP |
16, 10/22 |
Non-linear hyperbolic equations |
||
|
5, 9/3 |
ODE num. methods |
17, 10/24 |
Hyperbolic systems |
||
|
6, 9/5 |
Euler method analysis |
18, 10/29 |
Finite volume |
||
|
7, 9/10 |
Stability, consistency |
19, 10/31 |
Splitting methods |
||
|
8, 9/12 |
Bnd. Loc., convergence |
11/5 |
Lab: Spectral differentiation |
|
|
|
9, 9/17 |
Fourier analysis |
20, 11/7 |
Spectral methods |
||
|
10, 9/19 |
1D Heat eq. FDM's |
21, 11/12 |
Compact finite differences |
||
|
11, 9/24 |
2D Heat eq. |
22, 11/14 |
Finite element methods |
||
|
9/26 |
Lab: Fourier space |
11/19 |
Lab: 2D Navier-Stokes equations |
|
|
|
12, 10/1 |
Linear advection: characteristics |
23, 11/21 |
Finite element formulations |
||
|
13, 10/3 |
Basic advection methods |
24, 11/26 |
Ritz formulation example |
||
|
14, 10/8 |
Advection schemes: stability |
35, 12/3 |
Project presentations |
|
General bibliography
The following texts serve as general background material. Students are encouraged to actively read from these texts material related to the current lecture. Homework and final examination questions from this material should be expected.
Finite Difference Schemes and Partial Differential Equations, John Strikwerda
Numerical Methods for Conservation Laws, Randall LeVeque
Finite Difference Schemes for Computational Fluid Dynamics, Phillip Colella & Gerry Puckett
Suggested reading for specific topics
Material specific to a certain topic shall be listed here as the course progresses. Generally the books or articles will be placed on reserve at the Mathematics Library in Phillips for the time period indicated
P.G. Garabedian, Partial Differential Equations, Chapter 1, The Method of Power Series. (17 pp.). On reserve: Aug. 22 - Aug. 29. Questions to ponder: (1) Consider the relevance and applicability of the methods used in the Cauchy-Kowalewski theorem to practical computations; (2) Try to find a biography of Sophia Kowalewski (also transcribed as Sonja Kovalevski) and place the Cauchy-Kowalewski theorem in historical context.
G. Dahlquist, Numerical Methods, Prentice-Hall, 1974, Sections 7.1 (Difference operators), 7.2 (Richardson extrapolation), Chapter 8 (Differential equations)
Assignments
Homework assignments
|
Date issued |
Date due |
Topic |
Homework |
Solution |
|
8/20 |
8/29 |
Finite differences |
(too easy!) |
|
|
9/3 |
9/17 |
ODE's |
||
|
9/17 |
10/1 |
Heat equation |
||
|
10/1 |
10/15 |
Hyperbolic equations |
||
|
10/15 |
10/29 |
Non-linear hyperbolic eqs. |
||
|
10/29 |
11/12 |
.ps .pdf |
||
|
11/12 |
11/26 |
.ps .pdf |
.ps = Postscript File, .pdf = Portable Document File, .html = Web page, .nb = Mathematica notebook .m = Matlab m file .f90=Fortran 90 source file
Supplementary reading assignments
-
R. Courant, K. O. Friedrichs and H. Lewy, "Ueber die partiellen Differenzengleichungen der mathematischen Physik," Mathematische Annalen 100 (1928), 32-74. Translated as: "On the partial difference equations of mathematical physics," IBM Journal of Resarch and Development 11 (1967), 215-234. Students: Sarah Batey, Joohee Lee, Xiaoyu Zheng. To be presented to class on September 5.
-
Dahlquist, G. "A Special Stability Problem for Linear Multistep Methods". BIT, 3:27 (1963). I've checked extensively, but it seems that this classic is not available in the UNC libraries or at Duke. The basic material from the article is well covered in the book: Numerical Methods, Dahlquist, G. and Björck, Äke, sections 8.5.3, 8.5.4, Prentice-Hall, 1974. Students: Alison Hall, Joohee Li, Omar Tolbert. To be presented to class on September 12. Note: checking that the original article or some reprint is not available took some time, so the assignment was posted later than I would have liked. This is compensated by the fact that the material is closely related to the course work and is quite concise.
Computational work
It is essential that students acquire a basic familiarity with the process of developing and using scientific software. The homework assignments shall contain gradually more difficult problems that are intended to be solved using programs written by the students. The problems in the first homework assignments are small enough that they can solved using Matlab. Later problems are more computationally intensive and require writing programs in a compiler language such as C or Fortran. Students are encouraged to observe advanced techniques by attending the Scientific Computing Club (SCC) that shall meet during the Fall 2002 semester.
Fourier solution of common PDE's (09/26/02)
Here is a Matlab m-file script that computes solutions to an advection, heat, dispersive equation using Fourier analysis. The procedures closely mirror the analytical framework presented in the lecture notes.
Basic advection equation schemes (10/10/02)
Here is a Matlab m-file script that applies the upwind, Lax-Friedrichs and Lax-Wendroff schemes to the constant velocity advection equation.
Spectral differencing (11/05/02)
A small Matlab m-file showing how to compute derivatives spectrally.
2D Navier-Stokes equations (11/14/02)
A Matlab m-file to solve the 2D Navier-Stokes equations in the vorticity - stream function formulation.
Final project topics
Individual final project topics shall be listed here on November 5. Students must submit their choice of topic by November 12. Projects are due and shall be presented to class on December 3.
Final examination preparation
Questions similar to those students can expect on the final examination. .ps .pdf