Interactive Earth is a project that I have done on the side during my graduate studies in geodynamics at UC Berkeley. It is intended as a teaching tool for demonstrating what mantle convection is, and how it depends on Rayleigh number.
Research codes for mantle convection use modern and accurate numerical methods, and typically take hours, days, or weeks to run. Here the goal is speed over accuracy. I rely on a mishmash of different computational methods for solving the governing equations, where each method is very carefully chosen to be as fast as possible.
I solve the Stokes equation spectrally using the stream-function formulation. Fourier transforms are performed by the awesome FFTW library. I solve the diffusion equation with a dimension-split finite difference approximation, allowing it to be solved in O(N) operations. I solve the advection equation with a semi-Lagrangian approach, allowing me to take timesteps significantly larger than the CFL number.