For a semester project, consider the four topics outlined below. I have additional materials I will give you to help you out on both the mathematics and the computations.
in the unit square, travelling with velocity 
. A particle moves freely
unless it collides with another. In that case, a restoring force is generated on
both particles, composed of (1) a normal part proportional to the (virtual)
particle-particle overlap (2) a tangential part, a fraction of the normal
force (3) some dissipation. At the boundaries, one could either impose
periodicity, or (imperfect) reflection, or a prescribed velocity or force.
As the number of particles increases, detecting particle-particle interactions becomes
computationally expensive; more than a dozen or two and you can no longer afford to
simply check all potential pair interactions. How can you structure a calculation to
make a computation with a few hundred particles feasible? Code and run some interesting
problems. (If you already know something on this topic, you might
want to look into a Monte Carlo simulation instead.)
,
write a multigrid code and make sure it works when 
undergo large jumps
(e.g., 
.
(2) in 2D, write a 2-grid code that can solve 
,
with Dirichlet boundary conditions, and varying r. In both cases,
how would you incorporate Neumann data?
. That is, as 
vary, how do
solutions change?