Numerical Methods: Finite Differences - Environmental Modeling - page 369

here formulated for a function u depending on the independent variable x .
For second order derivatives the central FD scheme is usually without alterna-
tive. It is obtained by using the finite difference approximation of the first order FD
approximations:

2 u
@x 2

@ 1

D x

u ð x þ D x Þ u ð x Þ
D x

u ð x Þ u ð x D x Þ
D x

(21.10)

or (compare Sidebar 4.2)

2 u
@x 2

@ uðx þ D xÞ 2 uðxÞþuðx D xÞ

D x 2

(21.11)

A simple example may illustrate the entire procedure. FD transforms the differ-
ential equation for 1D steady state transport

D @

@x 2 v @ c

2 c

@x l c ¼

0 (21.12)

into

D c ð x þ D x Þ c ð x Þþ c ð x D x Þ
D x 2

v c ð x Þ c ð x D x Þ
D x

l cðxÞ¼ 0 (21.13)

Here the backward FD for the first order derivative has been chosen. In fact the
backward FD delivers a numerical algorithm that usually converges much better
method the version, in which forward or central schemes are used. For details about
this consult an introductory textbook on numerics (for example: Thomas 1995 ;
Moler 2004 ).
Equation ( 21.13 ) serves as basis for a numerical solution of ( 21.12 ) if we divide
the model region of interest, that is an interval x min x x max on the x -axis, into
peaces of length

D x . Using the numerical method we can obtain approximate values
for c ( x ) at the mesh positions, called nodes x

D x; ::::x max D x
(lets take x as a vector here). For each position x ( 21.13 ) states a relation between
the (unknown) function value c ( x ) and the (also unknown) function values at the
neighboring two nodes.
Boundary conditions have to be considered separately, but without problem.
A Dirichlet boundary condition at the left side of the interval for example,
cðx min Þ¼c 0 , according to ( 21.13 ) leads to:

¼ x min þ D x; x min þ 2

D cðx min þ

2

D xÞ

2 cðx min þ D xÞþc 0
D x 2

v c ð x min þ D x Þ c 0
D x

l cðx min þ D xÞ¼ 0

(21.14)