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In-Depth Information
k
x
k
y
k
0
0.5
0.5
1
0:193147
0:193147
2
0:043329
0:043329
3
0:001934
0:001934
3:75
10
6
3:75
10
6
4
1:40
10
11
1:40
10
11
5
We observe that, as in the scalar case, Newton's method gives very rapid conver-
gence toward the solution x
D
y
D
0.
4.6.5
The Nonlinear System Revisited
In Sect.
4.6.2
on page
127
we introduced the system
u
0
D
v
3
;
u
.0/
D
u
0
;
v
0
D
u
3
;
(4.169)
v
.0/
D
v
0
;
and the implicit Euler scheme
u
nC1
u
n
t
v
nC1
v
n
t
D
v
nC1
;
D
u
nC1
;
(4.170)
where we have used standard notation. The nonlinear system (
4.170
) can be rewrit-
tenontheform
u
nC1
C
t
v
nC1
u
n
D
0;
(4.171)
v
nC1
t
u
nC1
v
n
D
0:
We note that for a given time level t
n
D
nt ,the(
4.171
)definea2
2 system
of nonlinear algebraic equations. By setting ˛
D
u
n
, ˇ
D
v
n
, x
D
u
nC1
,and
y
D
v
nC1
, we can write the system in the form
f.x;y/
D
0;
g.x; y/
D
0;
(4.172)
where
f.x;y/
D
x
C
t y
3
˛;
g.x; y/
D
y
t x
3
ˇ:
(4.173)
We can now solve system (
4.172
) using Newton's method. We set x
0
D
˛, y
0
D
ˇ
and iterate as
