Transport - Transport and Fate of Chemicals in Soils: Principles and Applications - page 90

and

−

∂
∂

C
z

C C

+

i ++ + 1, j 1 ij , 1

v = v Oz () (3.61)

z

where O(Δ z ) 2 and O(Δ z ) are the error terms associated with the above
finite-difference approximations, respectively. In Equation 3.53, the second-
order derivative (the dispersion term) is expressed in an explicit-implicit
form commonly known as the Crank-Nickolson or central approximation
method (see Carnahan, Luther, and Wilkes, 1969). This is obtained using
Taylor series expansion and is based equally on time j (known) and time
j + 1 (unknown). Such an approximation has a truncation error, as obtained
from the Taylor series expansion, in the order of (Δ z ) 2 , which is expressed
here as O (Δ z ) 2 . In Equation 3.54, the convection term was expressed in a
fully implicit form, which resulted in a truncation error of O (Δ z ). In this
numerical approximation, for small values of Δ z and Δ t , these truncation
errors were assumed to be sufficiently small and were therefore ignored
in our analysis (see Henrici, 1962). Chaudhari (1971) showed that due to
numerical approximations, a correction to the dispersion term D is needed
such that

D =D+ D

*

n (3.62)

where D * is the corrected dispersion and D n is a numerical dispersion term that
is obtained from rearrangement of the higher-order terms of the Taylor series.
For derivatives based on central differences, D n is given by (Chaudhari, 1971):

v vt

R

= z −

D (3.63) n

2

Incorporation of D n is a simple task and can yield significant improvement
to numerical approximations.
The time-dependent term of Equation 3.24 was expressed as:

−

∂
∂

C
t

CC
t

+

ij , + 1 ij ,

R = Ot ()

R (3.64) ij ,

where the retardation term R was solved explicitly as:

=+ ρ

K f

R 1 b C

b − 1

(3.65)

Θ