GPUs are making this the era of high powered desktop computing and to prove the point what about a nice example of fluid simulation - video and code included.
The idea is to get close to the physics by solving the Navier-Stokes equation so the smoke you see swirling around the cylinder isn't just a special effect but a simulation of the fluid flow. Watch the video to see how it performs. It is worth mentioning that the video is a representation of the simulation in realtime - that is it is what you see when you run the program.
The code, which the author has placed in the public domain, is also available for Visual Studio 2010 and gcc. If you are interested in fluid simulation this is a good place to start.
Yes - smell. Diophantine equations are just polynomial equations that use nothing but integers for their coefficients and solutions. They are very hard to solve and often very important.
