Euler, Laplace and Archean

I received a friendly mail from John Hallam about my Archean project:

Off the top of my head, what you have described sounds like an Euler-method integrator for the laplace equation with 6 coupled potentials.  You should be able to get lots of interesting behaviour out of that.  If you make the local kernel even more flexible -- rather than just an average, which I think implements the laplace operator -- you could allow other local operators instead -- then you have a system that approximates the solution of other PDEs, for example the wave equation,

Unfortunately I'm not really sure what this all means. I can see how Archean is like systems that predict the spread of diseases, which I think the Eurler-method is used for. I would be interested if someone could point me towards more local operators.