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.