### Summary
This paper is an introduction to the Navier-Stokes equations and the methods proposed by the graphics community (up through Stam's Stable Fluids paper) for solving them.
### Problem
The Navier-Stokes equations were known to be a good approximation for fluid behavior, but solving them's a tricky problem, and it wasn't until Stam that there was a good, stable way to simulate them reasonably fast.
### Methods Used
Stam's method separates the Navier-Stokes equations into components and solves them individually, as follows
1.  At the beginning of the solution process, each cell's velocity is set to 0
2.  ...+f : Any external forces (gravity, magic, etc.) are added to the velocity
3.  ...-(u*del)u... : Using a fancy Semi-Lagrangian scheme, the velocity field is advected along itself, enforcing conservation of momentum
4.  ...+v*del^2(u)... : Stam solves for the updated velocity of this step using the velocity found in step 3 and an implicit solver.
5.  ...-(1/rho)*del(p) : This term isn't solved as such, but using the Helmholz-Hodge decomposition, Stam ensures that del(p) is 0 everywhere in the field by removing any divergence in the field's velocities
### Key Ideas
* Semi-Lagrangian advection
* Implicit solution using the conjugate gradient method of the viscosity term of the Navier-Stokes equations
* The Helmholz-Hodge decomposition for resolving divergence in the velocity field