Leaching of Contaminants to Groundwater - Dealing with Contaminated Sites: From Theory Towards Practical Application - page 829

relationships (Šimunek and Van Genuchten 2006 ). Numerical methods subdivide
the time and spatial coordinates into smaller pieces, such as finite differences, finite
elements, and/or finite volumes, and reformulate the continuous form of governing
partial differential equations in terms of a system of algebraic equations. In order
to obtain solutions at prescribed times, numerical methods generally require inter-
mediate simulations (time-stepping) between the initial condition and the points in
time for which the solution is needed.
Reviews of the history of development of various numerical techniques used in
vadose zone flow and contaminant transport models are given by Van Genuchten
and Šimunek ( 1996 ) and Šimunek ( 2005 ).

18.5.1.1 Finite Differences

Finite difference methods are generally very intuitive and relatively easy to imple-
ment. Time and space are divided into small increments z )

(Fig. 18.16 ). Temporal and spatial derivatives in the governing equations are then
replaced with finite differences (formally using Taylor series expansions). For exam-
ple, the standard advection-dispersion equation for steady-state water flow (Wang
and Anderson 1982 ) given by

t and z (or x and

2 c

∂ c

t =− ∂

J

D ∂ v ∂

c

z =

− (18.58)

z 2

∂ ∂ ∂ ∂ z

can be approximated as follows using an explicit (forward-in-time) finite difference
scheme:

J i + 1 / 2 − J i − 1 / 2

D c i + 1 −

2 c i c i − 1

v c i + 1 −

c i − 1

c j + 1
i

c i

+

−

=− = − (18.59)

z ) 2

t z ( 2 z

t i t i+ 1

x i -1 x i x i+ 1

z i+ 1

z j+ 1

c i- 1, j +1 c i , j +1 c i +1, j +1

j

c c

j +1
i +1 i +1

Δ z i

Δ z j

Δ z

Δ z

z j

z i c i- 1, j c i +1, j

c i , j

j

c c

j +1

Δ z j- 1

Δ z i- 1

z

z

z j- 1

c i +1, j −1

c i- 1, j- 1 c i , j- 1

z i- 1

j j + 1

c c

i − 1 i − 1

Δ x

Δ t

Δ x i -1 Δ x i

t x

Fig. 18.16 Examples of the spatial and temporal difference discretization of a one-dimensional
problem ( left ), and the finite difference discretization of a two-dimensional domain ( right )(after
Šimunek and Van Genuchten ( 2006 ))