Information Technology Reference
In-Depth Information
Fig. 7.12
Illustration of the
updating formula (7.91);
u
4
is
computed from
u
3
,
u
4
,and
u
5
t
t
3
Δ
t
t
2
t
1
t
0
x
0
x
1
x
2
x
3
x
4
x
5
x
6
x
7
x
8
x
9
x
Δ
x
new time level
1
s
s
previous time level
1−2s
t
x
Fig. 7.13
Illustration of the computational molecule corresponding to the finite difference scheme
(7.91). The weight s is equal to t =x
2
to the circles in Fig.
7.12
as a
computational molecule
, especially if we write the
weights inside the circles and connect the circles by lines, as depicted in Fig.
7.13
.
Because new
u
`C1
i
values can be computed explicitly from a simple formula,
where all the quantities are known, we refer to the finite difference scheme as
explicit
. The scheme is often referred to as the explicit Euler scheme or the for-
ward Euler scheme (the term “forward” relates to the forward difference in time).
The opposite type of schemes, the so-called
implicit
schemes, typically couple all
the new
u
`C1
i
values for i
D
0;:::;n in a linear system. This means that we need
to solve a linear system to find the
u
values at a new time level. Implicit schemes
hence imply a more complicated computational procedure, but implicit schemes
have some numerical advantages over explicit schemes. Section
7.5
is devoted to
implicit schemes for diffusion problems.
