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finite number of layers or grids as shown in Figure 3-5. Equation (16)
was rearranged so that unknowns were on the left side of the equation
and the knowns were on the right side such that
At K At
i-1/2 k+1
) h + ( 1 +
2k i-1 2
AZ C A Z
At
K
i-1/2
i+1/2
1
2 k
AZ C
i
) h
k+1
k k k
At K At K K At
i+1/2 k+1 k i-1/2 i+1/2
- ( ) h = h + ( ) S
2 k i+1 i k k i
az C az C C
i i i
(19)
Equation (19) was written for each of the interior grids in the soil
profile. The top and bottom grids of the system required equation (19)
to be modified so that the equation would describe the boundary
conditions. The system of equations which are formed by applying
equation (19) to each grid produces a tridiagonal matrix which is
implicit in terms of h, water potential. The tridiagonal matrix was
solved using Gauss elimination. From explicit linearization, the
coefficients of the unknown h terms were known at each time step. Thus
equation (19) was reduced to
k+1 k+1 k+1
Ah + B h + C h
i i-1 i i i i+1
= D
(20)
where
k
At K
i-1/2
A = ( )
i 2 k
A z C
(21)
1