Hello Joe,
The project I am contemplating is to try to use a Mimetic approach to solving some of the PDEs in ETK.
Can you point me to the places in the code where PDE solutions are being performed?
The main PDE (that is solved as a straightforward PDE) in the Einstein Toolkit are Einstein's equations of General Relativity which are solved by the McLachlan thorn ML_BSSN. That thorn is actually auto-generated from a Mathematica description of the PDE that you can find in the file arrangements/McLachlan/m/McLachlan_BSSN.m (or so).
A simpler equation would be the scalar wave equation
(-\partial^2_t + \Delta) \phi = 0
which is our usual test case for a hyperbolic equation (the ET is almost exclusively used for those).
There are any number of sample codes implementing this equation eg in the CactusExamples or CactusWave arrangements or also one for Kranc called SimpleWave.
There is also current project called NRPy+ by Zach Etienne to generate codes for PDEs using python (https://math.wvu.edu/~zetienne/SENR/), who is also subscribed to this list.
The other big set of PDEs in the ET are the equations of general relativistic (magneto-)hydrodynamics ie flux conservative equations of the form
\partial_t U + \partial_i F^i(U) = S(U)
which are highly nonlinear and prone to formming shocks. These are solved using high-resolution, shock capturing, finite volume methods in the thorns GRHydro and IllinoisGRMHD, providing two independent implementations of codes to simulate relativistic hydrodynamics.
Yours, Roland