Appendix C: FINITE DIFFERENCE FORM OF THE THIRD TYPE BOUNDARY CONDITION

As shown in equations (3.15) and (3.16) in the Section 3.2.1, the third type boundary condition at the top of the soil column is expressed as:

(C.1)

(C.2)

If we assume D is constant over the top pixels involved here, the finite difference form of the first term of the left hand of the above equations can be written (omitting the subscript w) as:

(C.3)                

If we assume qw is constant over top two pixels involved here, the second term of the equations (C.1) and (C.2) becomes

(C.4)                                         

By substituting the equations (C.3) and (C.4) into (C.1) and (C.2),

(C.5)                                  

where C0 becomes zero when t>t0.

When i = 1 (i.e., at the first cell), the equation (C.5) can be written as:

(C.6)                               

Note that the C0k+1 and C0k terms represent outside the column boundary. The same terms also can be found in the governing equation (3.36) of Section 3.3.1 by applying i = 1. When i = 1, the equation (3.36) becomes as follows.

(C.7)                      

Rewriting equation (C.7) and rearranging the Ck+1 terms into the right hand side and Ck terms into the left side of equation, the following can be derived.

(C.8)

where

Because we assumed D and qw are constant, M1' and N1' become 0. Then





Last modified: Oct 15, 1999
VG Model / Samuel Lee / VADOSE.NET