Information Technology Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
1
2
3
4
5
6
t
Fig. 4.8
The analytical solution .
u
D
cos.t /;
v
D
sin.t // and the numerical solution .
u
n
;
v
n
/,in
dashed lines
, produced by the implicit Euler scheme (
4.130
)
An implicit Euler scheme for this system reads
u
nC1
u
n
t
v
nC1
v
n
t
D
v
nC1
D
u
nC1
;
;
(4.142)
which can be rewritten in the form
u
nC1
C
t
v
nC1
u
n
D
0;
(4.143)
v
nC1
t
u
nC1
v
n
D
0:
Observe that in order to compute .
u
nC1
;
v
nC1
/ based on .
u
n
;
v
n
/, we need to solve a
nonlinear system of equations. To clarify matters, we study (
4.143
) for a fixed value
of n.Let˛
D
u
n
and ˇ
D
v
n
. We want to compute x
D
u
nC1
and y
D
v
nC1
.Let
f.x;y/
D
x
C
t y
3
˛;
g.x; y/
D
y
t x
3
ˇ:
(4.144)
Then system (
4.143
) can be written on the generic form
f.x;y/
D
0;
g.x; y/
D
0:
(4.145)
