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and
D
c
C
c
2
t
C
.t
2
/:
v
(4.19)
O
Here, (
4.18
) is an unreasonable solution because 1=t becomes infinite as t goes
to zero. The solution
v
behaves well, see (
4.19
). We therefore have, from (
4.15
),
that
1
p
1
4t c
2t
v
D
:
(4.20)
Based on this, we conclude that the implicit scheme
u
nC1
t
u
nC1
D
u
n
(4.21)
can be written in the computational form
1
p
1
4t
u
n
2t
u
nC1
D
:
(4.22)
To summarize, we have seen that the equation
v
t g.
v
/
D
c
(4.23)
can easily be solved analytically when
g.
v
/
D
v
(4.24)
or
g.
v
/
D
v
2
:
(4.25)
More generally, we can solve (
4.23
) for linear or quadratic functions g, i.e., func-
tions of the form
g.
v
/
D
˛
C
ˇ
v
C
v
2
:
(4.26)
But what happens if
g.
v
/
D
e
v
(4.27)
or
g.
v
/
D
sin.
v
/‹
(4.28)
