44
solution techniques utilize the finite difference form of the
differential equations. The differential operator in the continuous
domain becomes a difference operator in the discretized domain.
Difference operators are either forward, backward, or central
difference operators. The difference operators for the function of the
independent variable,
x, [ f(x)
] are
forward:
df
3x
o
III
f(*j+l) f(xj)
xj+l xj
(2-52)
backward:
3f
vxf -
f(Xj) f(Xj.!)
(2-53)
3x
xj XJ-1
central:
3f
35T
5xf
(2-54)
1 f(xj+l) f(xj) f(xj) f(Xj.!)
xj+l xj + xj xj-l
In expressing the differential equations in difference form, the
derivative with respect to time is accomplished using the forward
difference operator, while spatial derivatives are expressed using any
of the three difference operators. The continuous domain must first be
discretized or divided into several discrete regions such that the
finite difference approximation of the differential equation approaches
the differential equation in the limit of the grid spacing going toward
zero (Figure 2-5).
The partial differential equation defining the conservation of
energy within the soil profile (equation 2-26) stated that the change
in sensible heat over time is due to heat transferred by conduction
plus sensible heat carried by the diffusion of water in the liquid and