4.1. Finite Difference Methods
75
on. Thus. all first order and second order partial derivatives ofthe concentration at any node can be represented by linear combinations of the concentrati on values at that node and its neighboring nodes.
4.1.2 Finite Difference Solutions Jor One-Dimensional
Dispersion Problems
Consider the following one-dimensional advection-dispersion equation:
(4.1.9)
The time range [0, T] and space region [0, L] are both divided into a uniform
grid with time and space intervals Ilt and L\x, respectively. The concentration
value ofthe i-th node located at Xi and at time tn may be written as Ci,n' Now,
let us derive a finite difference equation at any internal node to replace Eq.
(4.1.9). The left-hand side of the equation is often approximated by the forward difTerence:
oC Ci,n+1 - Ci,n
iftIlt
(4.1.10)
On the right-hand side of Eq. (4.1.9), 02CjOX 2 is replaced by Eq. (4.1.6), and
oC/ox may be replaced by forward, backward, or central formulas, i.e., Eqs.
(4.1.3), (4.1.4) or (4.1.5). If tin Eq. (4.1.5) and Eq. (4.1.6) is replaced by t n , t n + l
or the middle time between t n and t n +1' three finite difTerence schemes can be
obtained, that is, the explicit, implicit and Crank-Nicolson schemes.
(1) The explicit difTerence equation of Eq. (4.1.9) is
Ci,n+l - Ci,n = D Ci+l,n - 2Ci,~ + Ci- l ,n _ VCi+l,n - Ci- l ,n (4.1.11)
Ilt
(L\x)
2L\x
From this equation, we can directly obtain
[ DL\t
VL\tJ
[2DIltJ
[DL\t
VL\tJ
Ci,n+1 = (L\X)2 + 2L\x Ci- l ,n + 1 - (L\X)2 Ci,n + (L\X)2 - 2L\x Ci+l,n'
(4.1.12)
Therefore, the concentration values of all nodes at time tn+ l can be calculated
directly without solving any equations, provided that their values at t n are
known.
(2) The implicit difTerence scheme of Eq. (4.1.9) is
Ci,n+1 - Ci,n = D Ci+l,n+l - 2Ci,n+1 + Ci- l ,n+l _ VCi+l,n+l - Ci- l ,n+l
Ilt
(L\X)2
2 L \ x '
(4.1.13)
where the unknown concentrations Ci- l ,n+1, Ci,n+1 and Ci+l ,n+l at tn+1 are
75
on. Thus. all first order and second order partial derivatives ofthe concentration at any node can be represented by linear combinations of the concentrati on values at that node and its neighboring nodes.
4.1.2 Finite Difference Solutions Jor One-Dimensional
Dispersion Problems
Consider the following one-dimensional advection-dispersion equation:
(4.1.9)
The time range [0, T] and space region [0, L] are both divided into a uniform
grid with time and space intervals Ilt and L\x, respectively. The concentration
value ofthe i-th node located at Xi and at time tn may be written as Ci,n' Now,
let us derive a finite difference equation at any internal node to replace Eq.
(4.1.9). The left-hand side of the equation is often approximated by the forward difTerence:
oC Ci,n+1 - Ci,n
iftIlt
(4.1.10)
On the right-hand side of Eq. (4.1.9), 02CjOX 2 is replaced by Eq. (4.1.6), and
oC/ox may be replaced by forward, backward, or central formulas, i.e., Eqs.
(4.1.3), (4.1.4) or (4.1.5). If tin Eq. (4.1.5) and Eq. (4.1.6) is replaced by t n , t n + l
or the middle time between t n and t n +1' three finite difTerence schemes can be
obtained, that is, the explicit, implicit and Crank-Nicolson schemes.
(1) The explicit difTerence equation of Eq. (4.1.9) is
Ci,n+l - Ci,n = D Ci+l,n - 2Ci,~ + Ci- l ,n _ VCi+l,n - Ci- l ,n (4.1.11)
Ilt
(L\x)
2L\x
From this equation, we can directly obtain
[ DL\t
VL\tJ
[2DIltJ
[DL\t
VL\tJ
Ci,n+1 = (L\X)2 + 2L\x Ci- l ,n + 1 - (L\X)2 Ci,n + (L\X)2 - 2L\x Ci+l,n'
(4.1.12)
Therefore, the concentration values of all nodes at time tn+ l can be calculated
directly without solving any equations, provided that their values at t n are
known.
(2) The implicit difTerence scheme of Eq. (4.1.9) is
Ci,n+1 - Ci,n = D Ci+l,n+l - 2Ci,n+1 + Ci- l ,n+l _ VCi+l,n+l - Ci- l ,n+l
Ilt
(L\X)2
2 L \ x '
(4.1.13)
where the unknown concentrations Ci- l ,n+1, Ci,n+1 and Ci+l ,n+l at tn+1 are