What is the right side simulation supposed to represent with respect to the left side one?
I think the simulation works by tracking a bunch of particles (shown on the right) and then reconstructing an image by interpolating the particles (on the left). If you track smoke or a fluid using particles you might want the resulting image to still look smooth. I'm not sure what algorithms are used to do this, though.
From lecture I think it was that the motion of smoke can be described with a bunch of rings floating around. So that if you can, instead of keeping track of each particle of smoke, you can solve for the ring's position and then add the smoke into the scene.
@BellaJ It's a little hard to see from the picture, but the left picture is just some particles that track the motion of the smoke, whereas the right picture also shows a collection of polygonal curves that describe the physics of the smoke (namely, the distribution vorticity, which is something like the amount by which the smoke is "swirling around"). You can find the original paper here.
@HelloWorld, @jkalapos As I mentioned to @BellaJ, you can find all the details of how this algorithm works here.