The project work is not compulsory, but strongly recommended, and will count for 20% of your final mark. The deadline to hand in the project work is the 31th March (midnight). The presentation of the project will be on the 2nd/3rd of April.

This project aims at developing experience in implementing and experimenting with finite difference discretizations of PDEs. This is a very important goal of this course.

The project is a group assignment, and we encourage to work in the same groups as have already been established. If there are any changes, notice Sølve as soon as possible.

Your task is to choose a PDE and find out as much as you can about the problem. Then propose a finite difference discretization of the equation, and implement your method as efficiently as possible. A List of PDEs to choose from has been provided - if there is an equation you are desperate to study that is not on this list we may be able to accommodate this, let us know.

You are supposed to suggest a complete discretization scheme. Using the method of lines for time-dependent problems will reduce the number of points given. Using MOL can still be useful as an intermediate step to the full scheme, and also for creating reference solution for convergence plots.

Most of the problems have an intended real world application - we hope that this can help make the problems more approachable and interesting, and also reflects that most PDEs are studied with some applications in mind. We encourage you not to lose sleep about modelling matters, but it will be a nice bonus if the plots you display give an indication of what the equation is representing.

The project aims to encourage self-sufficiency and creativity by its open nature - we have not provided an enormous amount of information, and students are expected to do their own research into the problems. We are aware that this may be intimidating and will be available to provide guidance, particularly in the early stages of the project.

1. Scientifically written report
   • A description of your problem of choice, including a complete set of equations: the PDE(s), domain of integration, boundary conditions, and parameters used. This is also the place to say something about what your equation describes, and whenever relevant, particular expected behaviour.
   • The difference scheme used in the simulation, and a motivation for why this was chosen.
   • Presentation, verification and discussion of the numerical results. Verification can e.g. be a convergence plot, discussion will typically be vs. expected behaviour of the solution.
   • In the conclusion section it should be clearly stated who did what. It is important to note that all participants in a group share equal responsibility for everything the group hands in. Every participant of a group should know and understand everything the group has done. If the members of the group agree that the work has been distributed evenly, simply write that each of the group members has contributed equally to the project work.
   • The maximum number of pages is 8.
   • Use an appendix if you have some extra material you absolutely need to attach to the report, for example extra pictures or pseudo-code (but be advised that we will probably not read through the appendix)

1. Code files
   • You can send multiple files BUT it should be easy enough for us to run the code (make a README file with the instructions of how to run the code).
   • The code will be used if we need to check parts of the report that we do not understand and in order to see how efficient your implementation is.

Hand-in of the project will be through ovsys.

• The presentations will be on Tuesday April 2 and Wednesday April 3. Both the lecture times and the tutorial times will be used, so a total of 3 sessions.
• The schedule will be presented on this webpage on Monday April 1.
• Each group has 10 minutes for their presentation.
• All the group members shall participate.
• The presentations are open for all students in the class, but you are only obliged to be present for all the presentations in your own session.
• Aim at making the presentation interesting for your fellow students, who do not know anything about the equation you have chosen.

You can get a total of 20 points. The evaluation will take into account the degree of difficulty of the problem you have considered.

The presentation will be an important step in the evaluation process. This is mandatory. If you are sick, you should provide a medical certificate to Stian at the office of our Department and inform us. We will arrange an oral discussion of your project work at a different date.

From next week, there will be supervision in the computer lab on Wednesdays 10:15-12:00 and 16:15-18:00 (no lecture).

Additional supervision (subject to change):

If the time given is inconvenient for your group, please try to trade places with another group by yourself.

The presentation can be in English or Norwegian, of your own preference. English slides combined with a presentation spoken in Norwegian is also allowed.

1. Group 1 (Tumor invasion model)
2. Group 2 (Shallow water equations)
3. Group 4 (Non-linear Schrödinger equation)
4. Group 5 (Korteweg–de Vries equation)
5. Group 6 (Perona–Malik equation)
6. Group 7 (Gray–Scott equations)
7. Group 11 (Boussinesq equation)
8. Group 12 (Porous medium equation)

1. Group 3 (Tumor invasion model)
2. Group 8 (Shallow water equations)
3. Group 15 (Korteweg–de Vries equation)
4. Group 17 (Chaplygin equation)
5. Group 18 (Iceberg equation)
6. Group 19 (Allen–Cahn equation)
7. Group 20 (Perona–Malik equation)

1. Group 10 (Shallow water equations)
2. Group 13 (Perona–Malik equation)
3. Group 14 (Porous medium equation)
4. Group 21 (Helmholtz equation)
5. Group 22 (Biharmonic equation)
6. Group 23 (One-way wave equation)

• Release of project, formation of groups: 4th of March.
• Deadline for handing in the final report: 31th of March.
• Oral presentation: 2nd/3rd April.