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Based on these computations we expect that
(a) The analytical solution .F .t /; S.t // forms a circle in the F -S coordinate system
(b) A good numerical method generates values .F
n
;S
n
/ that are placed almost
exactly on a circle, and they get closer to a circle when t is smaller
We will look a bit closer at these two conjectures.
3.4.3
Circles in the Analytical Phase Plane
Our hypothesis, based on the computations above, is that the solution forms a circle
that seems to have the point (1,1) as its center. In order to see whether this is correct,
we define the function
r.t/
D
.F .t /
1/
2
C
.S.t /
1/
2
:
(3.26)
In Fig.
3.6
we have plotted an approximation to this function given by
r
n
D
.F
n
1/
2
C
.S
n
1/
2
(3.27)
1
0.9
0.8
0.7
0.6
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
t
Fig. 3.6
The r.t/ function, defined by (
3.26
), of the numerical solution for (
3.24
), which is
produced by the explicit scheme (
3.25
)usingt
D
1=1;000, F
0
D
0:9,andS
0
D
0:1