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so
.F .t /
1/
2
C
.S.t /
1/
2
D
.F
0
1/
2
C
.S
0
1/
2
(3.34)
for all t
0. This is just another way of deriving the fact that
r.t/
D
.F .t /
1/
2
C
.S.t /
1/
2
is constant.
3.4.5
Circles in the Numerical Phase Plane
We still consider the simplified model
F
0
.t /
D
1
S.t/;
F.0/
D
F
0
;
(3.35)
S
0
.t /
D
F.t/
1;
S.0/
D
S
0
:
We have seen above that for this system the function
r.t/
D
.S.t /
1/
2
C
.F .t /
1/
2
is constant for all t>0. An explicit numerical approximation of (
3.35
)isgivenby
F
nC1
D
F
n
C
t.1
S
n
/;
(3.36)
S
nC1
D
S
n
C
t.F
n
1/;
where F
0
and S
0
are given. In Fig.
3.6
above we saw that the discrete function
r
n
D
.F
n
1/
2
C
.S
n
1/
2
was almost constant, i.e., r
n
r
0
for all n
0.
Let us look a bit closer at this. We have r.t/
D
r
0
, and thus a perfect numerical
scheme should produce r
n
D
r
0
for all n
0. Now, of course, the scheme only
provides approximations, but this property can give some insight into how accurate
the approximation is. This is a general observation: If we have a property of the
analytical solution that is computable, this can be used to strengthen, or perhaps
weaken, our faith in the numerical solution. More specifically, we can use such a
property to compare different numerical approximations.
But let us do some more computations and regard r
n
r
0
as a measure of the
error in our computations. Suppose we want to solve system (
3.35
) from t
D
0 to
t
D
10. We choose t
D
10=N , and use N
D
10
k
for k
D
2; 3; 4; 5.InTable
3.1
we study
r
N
r
0
r
0
for these values of t .
