Numerical Solutions of Differential Equations
News
- The re-exam will be on Aug 23rd, at 13.00-18.00, in room 1537.
- On Thursday, June 14th at 10.00 in D41, the author of your course book, Randall LeVeque himself,
will come to KTH and give a talk on a topic (and in a room) that should be familiar to you, namely
hi-resolution finite volume methods for the shallow water equations. The application
that he is interested in is tsunami waves and he will e.g. show simulations of the devastating waves
triggered by the large earthquake in Japan last year. An abstract of the talk is here.
Welcome!
- Homework 4 is now corrected. Results have been emailed to you.
The Homework 4 report and older reports can be picked up at
the CSC school's student expedition (bottom floor, Osquars Backe 2).
- I have added another example exam from 2007 below, with suggested
solutions included. I would also like reiterate
that the example exams do not cover precisely the same things that
we have covered this year.
Therefore some topics that show
up in the old exams
will not show up on this year's exam, for instance:
- Spectral methods
- Multigrid
- Stiff source terms and splitting
- Proof of existence of solutions
- von Neumann analysis for systems of equations
- Riemann problems for systems of equations
- Stability for nonlinear equations and contraction mappings
On the other hand, there may be questions on Helmholtz and high frequency approximations.
- The complete
Lecture notes 5 are now online
- HW5 is now online
- Final deadline for supplementing HW 1-4 are on May 16th
- Lecture at 15-17 on Tue May 8 is replaced by a lab session in Grön as usual
- Lab at 15-17 on Thu May 3 is CANCELED
- Homework 3 is corrected. Here are some
comments from the grader
(Jesper Oppelstrup).
- Homework 4 is now up.
- Lecture 12 has been moved to Monday April 23, same hour and place. (Used to be Tue April 24.)
- The missing Lecture notes 3 are now posted below. (Some cosmetic updates of other lecture notes too.)
- Rooms are now decided for the rest of course. We will be
in 1537 which is located in the main CSC building (Lindstedsvägen 3) on
level 5.
- Lecture 9 has been moved to Monday March 26 at 15-17. (Used to be March 27, same time.)
Instructor
Responsible for the course and lecturer:
Olof Runborg,
olofr@nada.kth.se.
Literature
Homework
The homework sets are:
- Homework 1 -- Check the
Demmel notes
for more about the matrix relation in 2.3 and
the Kronecker products in 2.4.
For the convergence parts, check the new
notes.
Deadline for Homework 1 is Monday 13 Feb
- Homework 2 -- Reading in Leveque:
Chaps. 2.7, 4-4.6, 7, 7.3.3, 13.1
Deadline for Homework 2 is Monday 27 Feb
- Homework 3 -- Reading:
Lecture notes 1 (updated!),
Lecture notes 4,
Leveque:
Chap. 8.
Grader's comments
on HW3.
Deadline for Homework 3 is Monday 2 April
- Homework 4 -- Reading in
Leveque: primarily Chaps 6 and 15.3-4 plus background in Chaps 11, 4.10-14, 12-12.1, 13. For the curious here are some
comments and discussions about exercise 2.2
Deadline for Homework 4 is Wednesday 2 May (OBS! Moved up because of Valborg.)
- Homework 5 -- Reading:
Lecture notes 5
Deadline for Homework 5 is Wednesday 24 May
How to hand in reports
Reports MUST have a cover page with author names, course code (DN2255) and homework number, and instructor name.
The reports can be handed in by
- e-mailing a link to Olof where your ps- or pdf-files can be downloaded, or
- e-mailing Olof that a paper version has been deposited in
(a) the Nada/CSC mailbox, bottom floor, Osquars B 2, or
(b) Olof's mail stop (floor 4, Osquars B.2)
- For HW1: Also email the m-files for your code.
Obs! Please do not email your full reports! This will rapidly fill up my mailbox.
For plots in your report, make sure you
- label axes correctly
- explain what the plots show
- combine plots so that comparisons are easy.
MERE SHEAVES OF MATLAB PLOTS, ONE TO A PAGE, WILL NOT BE ACCEPTED.
Matlab
The exercises will require Matlab
programming. Students are expected to have procured
familiarity with this environment in previous courses. We will try to
emphasize the importance of writing well structured and efficient
Matlab programs in this course. Here are some notes on that topic:
Here are some reading tips for students who feel less comfortable with
the Matlab environment:
- Chapman, "MATLAB Programming for Engineers", CL-Engineering 2007
- Moler, "Numerical Computing with Matlab", SIAM 2004
Lecture and Computer Lab Schedule
The course has 14 lectures, and 8 supervised computer exercise classes,
scheduled according to the list below.
|
Week
|
Date |
Time
|
Class
|
Room
|
Topics
|
Reading
|
| 3 |
Wed 18 Jan |
10:00-12:00 |
Lecture 1 |
B24 |
Introduction to PDEs |
Lecture notes 1, Leveque 2-2.6 |
| 4 |
Wed 25 Jan |
10:00-12:00 |
Lecture 2 |
B24 |
Heat equation, well-posedness, properties, discretization |
Lecture notes 2, Leveque 4-4.2 |
| |
Wed 25 Jan |
13:00-15:00 |
Lab 1 |
Grön |
Homework 1 |
|
| 5 |
Tue 31 Jan |
10:00-12:00 |
Lab 2 |
Grön |
Homework 1 |
|
| |
Wed 1 Feb |
10:00-12:00 |
Lecture 3 |
B24 |
Heat equation, discretization, properties of discretization, Matlab hints |
Lecture notes 3,
Matlab notes (Lindbo, Getreuer) |
| 6 |
Tue 7 Feb |
10:00-12:00 |
Lab 3 |
Grön |
Homework 1 |
|
| |
Wed 8 Feb |
10:00-12:00 |
Lecture 4 |
B24 |
Hyperbolic equations, intro, characteristics |
Leveque 2.7-2.11 (2.12-2.14), 3-3.6, 9-9.1 |
| 7 |
Tue 14 Feb |
10:00-12:00 |
Lab 4 |
Grön |
Homework 2 |
Leveque 2.7, 4-4.6, 13.1 |
| |
Wed 15 Feb |
10:00-12:00 |
Lecture 5 |
B24 |
Hyperbolic equations cont.,
discontinuous solutions, boundary conditions,
basic discretizations |
Leveque 3.7-3.11, 4-4.1, 4.5-4.9, 7 |
| 8 |
Tue 21 Feb |
08:00-10:00 |
Lab 5 |
Grön |
Homework 2 |
|
| |
Wed 22 Feb |
10:00-12:00 |
Lecture 6 |
B24 |
Convergence theory for linear methods |
Leveque 4.3-4.4, 8-8.3.4, Lecture notes 4 |
| 9 |
Wed 29 Feb |
10:00-12:00 |
Lecture 7 |
B24 |
Convergence theory for linear methods cont.,
errors for discontinuous solutions |
Leveque 8.6-8.7, Lecture notes 4 |
| 12 |
Tue 20 Mar |
15:00-17:00 |
Lecture 8 |
1537 |
Nonlinear hyperbolic equations |
Leveque 11-11.13 |
| 13 |
Mon 26 Mar OBS!!! Changed date |
15:00-17:00 |
Lecture 9 |
1537 |
Finite volume methods for nonlinear hyperbolic equations, Godunov, Riemann problems, approximate Riemann solvers |
Leveque 3.8-3.9, 4.10-4.14, 12-12.1, 12.9, (13.2-13.12 no details), 15.1, 15.3-15.3.3 |
| |
Thu 29 Mar |
15:00-17:00 |
Lab 6 |
Grön |
Homework 3 |
|
| 15 |
Tue 10 Apr |
15:00-17:00 |
Lecture 10 |
1537 |
High resolution methods, total variation, slope and flux limiters |
Leveque 6-6.15, 12.8, 12.10, 15.4 |
| 16 |
Tue 17 Apr |
15:00-17:00 |
Lecture 11 |
1537 |
Helmholtz equation, intro |
Lecture notes 5 |
| |
Thu 19 Apr |
15:00-17:00 |
Lab 7 |
Grön |
Homework 4 |
|
| 17 |
Tue 23 Apr OBS!!! Changed date |
15:00-17:00 |
Lecture 12 |
1537 |
Helmholtz, cont. |
Lecture notes 5 |
| 18 |
Wed 2 May |
15:00-17:00 |
Lecture 13 |
1537 |
High frequency approximations, eikonal equation |
Lecture notes 5 |
| |
Thu 3 May |
15:00-17:00 |
CANCELED |
CANCELED |
CANCELED |
|
| 19 |
Tue 8 May |
15:00-17:00 |
Lab 8 |
Grön |
Homework 5 |
OBS! Lecture replaced by lab.
|
| 22 |
Thu 31 May |
14:00-19:00 |
Exam |
D41 |
All of the above |
All of the above, except the Matlab notes. |
Examination
To pass the course the student must
- Pass a written examination (3 ECTS).
The examination is with closed books. It consists of ca 5 questions that can give a maximum of 40 points. A total of 20 points guarantees a pass.
-
Solve the five homework problems (4.5 ECTS).
For each homework set a written report of good quality should be handed in. The students are encouraged to work in groups of two. The homework is graded and each set gives a maximum of 3-5 points, in total at most 21 points for all five sets. A homework problem is considered as "solved" if
at least half of the maximum points are given.
Final grade
The final grade of the course is based on the results of the homework (60%) and the written examination (40%). More precisely, the final score is between 0 and 100, given by the formula:
Score = Exam points + Homework points * 60/21.
The preliminary grading scale is
| Score | 0-49 | 50-59 | 60-69 | 70-79 | 80-89 | 90-100 |
|---|
| Grade | F | E | D | C | B | A |
|---|
(Subject to minor modifications.)
Exam hints
Exam questions will cover the reading material (see table above) as well
as the homework material. There will be no exam questions on Matlab, however.
The style of the exam will be as earlier years, exemplified by:
See also the following
list of exam questions
from 2010. (OBS! There is no guarantee that four out of five questions
are from this list this year!)
Note that the contents of the course have shifted slightly over
the years, so unlike in some of the old exams,
there will not be questions on e.g. spectral methods,
multigrid and stiff source terms. On the other hand, there
may be questions on Helmholtz and high frequency approximations.
Course activation in Rapp
For registration to the course, please contact your "kansli"/"studievägledning"/coordinater.
Once registered, for us to be able to repot your results on the course you
need to show that you are active in our reporting system Rapp.
This you can do by connecting to
Rapp-server.
Click on Logga in / Log in.
You will be forwarded to KTH's login page where you can log in with
your usual KTH username/password.
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