Cactus Code Thorn MaxwellToyCarpetX
Author(s)    : Erik Schnetter 
Maintainer(s): Erik Schnetter 
Licence      : LGPL
--------------------------------------------------------------------------

1. Purpose

Solve the Maxwell equations.



2. Derivatives, and constraint preservation

We store variables and calculate derivatives in a way that is very
similar to discrete differential forms. Scalars live on vertices,
vectors on edges, fluxes on faces, and densities in cells. This
happens in 4D, leading to a staggered evolution scheme. We have the
following variables:

- electric potential phi: vertex, staggered in time (equivalent to a
  timelike edge)

- magnetic vector potential A^i: (spacelike) edges

- electric field E^i: (spacelike) edges, staggered in time

- magnetic field B^i: faces

- electric charge density rho: vertex (equivalent to dual cell),
  staggered in time

- electric current density j^i: edges (equivalent to dual face)

- electric constraint div E: vertices (equivalent to dual cell),
  staggered in time

- magnetic constraint div B: cell

When calculating derivatives, we divide by the grid spacing (unlike
from differential forms). This is for convenience only.

In terms of differential forms, our variables correspond to the
following. The notation [phi, A] means that phi and A^i are the time
and space components of a 4-vector.

- A: [phi, A]: 1-form
- F: [E^i, B^i]: 2-form
- j: *[rho, j^i]: 3-form living on dual grid
- *[div E]: 4-form living on dual grid
- *[div B]: 4-form

4d Equations:

- F = dA
- dF = 0
- d*F = 0

3+1 Equations:

- E = dphi
- B = dA
- *d*E = *rho
- *d*B = *j
(TODO: Correct and complete equations)



3. How to calculate consistent derivatives with mesh refinement

Prolongation opterators need to be conservative in cell centred
directions, and interpolating in vertex centred directions. We want
linear accuracy for vertex centred and constant accuracy for cell
centred directions.