Establish a finite difference grid on the xy-plane with x grid points
denoted by
and y grid points by 
.
The grid spacing in the x-direction is denoted by hx and in
the y-direction by hy.
As in the one dimensional case, control points are established midway
between the x and y grid points.
Using the shaded control region shown in Figure 2,
we write a conservation law integral equation similar to Eq. 10.
Namely,
Figure 2: Control region for two dimensional material balance.
Equation 24
states that the change in M-content in the control region
equals the net flux of M across the boundary from 
to 
.
If we approximate all the spatial integrals in Eq. 24
using the midpoint
quadrature rule, then Eq. 24 yields
The time integrals can be handled as before. If we use a left hand quadrature rule, an explicit FDE results. The right hand quadrature rule gives the backward in time implicit FDE, and the midpoint rule the Crank--Nicolson FDE.
