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50% of your grade for the course will be based on completion of a research project.
This project should involve reading one or more relevant articles or book chapters (beyond the regular reading for the course), and typically will include development of a basic software implementation. One course session will be devoted to a discussion of a paper from your project; you will lead this discussion. To finalize the project, you have the option of writing a report or giving a presentation to the class.

Important project dates:

  • Project proposal due: February 28th
  • Progress report due: April 25th
  • Project presentations/reports due: May 5th & 8th

Progress report (due April 25th)

This is a written report of no more than 4 pages, meant to help ensure that you're making adequate progress on the project. You should explain ideas that you have learned so far, and describe any modifications you've had to make to the original project plan and scope. It's also a good idea to include some preliminary numerical results (since you have hopefully made at least a preliminary implementation of any algorithms involved).

Presentation (last week of class)

You will give a presentation to the class which should take 25 minutes, leaving 5 minutes for questions. The objective of the presentation is to teach the topic of your project to the other students in the course. In most cases, you should include computational results from your own code as illustrations or examples.

You are encouraged to come up with your own project topic. Guidelines for the proposal will be distributed during the first week of the course. Your project topic should not be very closely related to your thesis research.

Some potential project topics

Applications:

  • Tsunami or flood modeling - The shallow water equations are frequently used in both
  • Waves in periodic or random materials
  • Detonation waves - the mathematics of explosions
  • Magnetohydrodynamics - Magnetized ideal gases, important in many astrophysical applications as well as in fusion experiments
  • Compressible fluid dynamics applications (e.g. astrophysics)

Theory:

  • Glimm's random choice method - see Serre vol. 1, Ch. 5
  • Non-conservative hyperbolic systems - see FVMHP Section 16.5
  • Non-convex hyperbolic systems - see FVMHP Section 16.1
  • Nonlinear geometric optics - see Serre vol. 2, Ch. 11
  • Relaxation systems - see FVMHP Sections 17.17-17.18
  • Compensated compactness - a method for proving global (in time) existence of solutions; see Serre vol. 2, Ch. 9
  • Oleinik's entropy condition - see FVMHP Section 11.13

Numerics:

  • Discontinuous Galerkin methods - finite element methods based on piecewise continuous function spaces
  • Problems with spatially varying fluxes - see FVMHP Section 16.4 and RPJS
  • Well-balanced numerical methods - for solving problems with source terms that are near a steady state
  • Time integration for hyperbolic PDEs - efficiency, stability, accuracy, storage issues
  • Comparison of multidimensional algorithms in Clawpack - do transverse Riemann solves really pay off? See FVMHP Chaps. 20-21
  • Adaptive Mesh Refinement - to resolve fine structures without using a fine grid everywhere
  • Instabilities near stationary shocks (the “carbuncle” phenomenon)