Finite Difference Equations

Even though each of the equations only involves first order derivatives, simple backward or forward differences are not suitable for integration of three dimensional scalar fields. Gauss-Seidel requires a diagonally-dominant system for assured convergence. Though forward and backward finite differences will generally converge, a simple combination of both schemes will reduce the magnitude of the off-diagonal terms and thus guarantee convergence. In addition, this provides a set of difference equations which are symmetric with respect to grid traversal directions. First, some notational conventions:

The photon flux equation has the following finite difference relationships:

or,

Similarly, the ion production equation can be expressed as a finite difference equation.

So that,

And finally,

The free boundaries of the system (top and bottom ion boundaries and the daylight bottom and nighttime top flux boundaries) can easily be handled by extrapolating to virtual grid points beyond the boundaries and using the same difference equations. The virtual points, of course, are not integrated.