Numerical Methods: Finite Differences - Environmental Modeling - page 370

A Neumann condition at the right hand side, @c=@x ¼ 0, transferred to the first

order FD leads to:

D cðx max

2 D xÞcðx max D xÞ

D x 2

v c ð x max D x Þ c ð x max

2 D xÞ

l cðx max D xÞ¼ 0
(21.15)

D x

In that way we obtain as many equations as there are unknowns in the vector x.
Before we proceed with the description of the numerical procedure lets take a
partial differential equation as another example. Using the approximations ( 21.11 )
one obtains for the 2D Laplace equation in x , y -coordinates:

2 u 2 u

@y 2

@

@x 2 þ @

uðx þ D x; yÞ 2 uðx; yÞþuðx D x; yÞ

D x 2

(21.16)

uðx; y þ D yÞ 2 uðx; yÞþuðx; y D yÞ

D y 2

þ ¼ 0

which can be applied In 2D space domains, which are discretized into a rectangular
mesh or grid , as shown in the following figures.
At nodes at the boundary the discretization ( 21.16 ) changes according to the
boundary condition. Let's give one example for a Dirichlet boundary condition. If
at the right boundary we have a value u bnd specified, then we obtain instead:

u bnd 2 uðx; yÞþuðx D x; yÞ

D x 2

uðx; y þ D yÞ 2 uðx; yÞþuðx; y D yÞ

D y 2

þ ¼ 0

(21.17)

Other node values have to be replaced, if the Dirichlet condition is for a node on
the other sides. For a Neumann condition at the right side, we have to make the
following modification:

u ð x ; y Þþ u ð x D x ; y Þ
D x 2

uðx; y þ D yÞ 2 uðx; yÞþuðx; y D yÞ

D y 2

þ ¼ 0 (21.18)

uðxþ D x;yÞuðx;yÞ
D x

¼ @u

This results from the first order approximation 0

@x

(see ( 21.9 )), which is equivalent to uðx þ D x; yÞuðx; yÞ¼ 0(Fig. 21.2 ).

In case of an equidistant mesh with equal grid spacing h ¼ D x ¼ D y the approxi-
mation of ( 21.16 ) simplifies to the well-known five-point stencil:

1
h 2 uðx þ D x; yÞþuðx D x; yÞþuðx; y þ D yÞþuðx; y D yÞ

ð 4 uðx; yÞ Þ ¼ 0

(21.19)