One of the first equations I discretized and simulated was the wave equation using a finite difference scheme. I ended up with a simulation, which is quite pretty:

The biggest PDE project I have done so far was with the Leuven iGem-Team. We simulated the behavior of pattern forming bacteria using pure PDE and a PDE-Agent hybrid models. The pure PDE simulations have been generated by discretizing a modified Keller-Segel system of equations using a finite volume method in Matlab:

In order to take cell adhesion into account we created this hybrid model:

The international jury in Boston nominated us for the best model award, which means they saw us in the top 5 of overgrad teams in the modeling category.

Interested? Read more here:

Leuven iGem 2015 wiki