Civil Engineering Reference
In-Depth Information
§ 3. Finite Difference Methods
The finite difference method for the numerical solution of an elliptic partial differ-
ential equation involves computing approximate values for the solution at points
on a rectangular grid. To compute these values, derivatives are replaced by di-
vided differences. The stability of the method follows from a discrete analog of
the maximum principle, which we will call the
discrete maximum principle
.For
simplicity, we assume that
is a domain in
2
.
R
Discretization
The first step in the discretization is to put a two-dimensional grid over the domain
. For simplicity, we restrict ourselves to a grid with constant mesh size
h
in both
variables; see Fig. 2:
h
:
={
(x, y)
∈
;
x
=
kh, y
=
h
with
k,
∈ Z}
,
∂
h
:
={
(x, y)
∈
∂
;
x
=
kh
or
y
=
h
with
k,
∈ Z}
.
We want to compute approximations to the values of
u
on
h
. These approximate
values define a function
U
on
h
∪
∂
h
. We can think of
U
as a vector of dimension
equal to the number of grid points.
Fig. 2.
A grid on a domain
We get an equation at each point
z
i
=
(x
i
,y
i
)
of
h
by evaluating the
differential equation
Lu
=
f
, after replacing the derivatives in the representation
(2.4) by divided differences. We choose the center of the divided difference to
be the grid point of interest, and mark the neighboring points with subscripts
indicating their direction relative to the center (see Fig. 3).
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