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.