Types of N-body ODE Integration Methods
June 9, 1996
37
2.4 Other formulas (Mid-point method) (none are implemented)
There turns out to be many other formulas of the form:
where the
n
's and
m
's may reference either into the future or into the past. Several of these
have interesting properties.
One of the more useful formula is the mid-point formula:
This formula is no more expensive to calculate than Euler's method, but it is of
O(h
2
)
instead of
O(h)
. The mid-point method is named because it is saying that a good
estimate for the average acceleration is at the middle of the time period.
The down side of the mid-point formula is that it is not very stable. The locations
of the even time periods
(t, t+2h, t+4h)
depend only on the other even time periods and
they are only loosely coupled with the odd time periods via the slopes
1
. (The same is true
for the odd time periods.) So, it is easy for the time periods to get out of sync and for the
even and odd time period points to start moving along totally separate paths. For this
reason, it is not recommended to use the mid-point method for a large number of steps.
Sources: (1:320-5,4:375-6,14:580)
1. This method is also known as the "leapfrog" method because of the way even and odd time periods leap-
frog over each other.
x t h
+
(
)
a
1
x t n
1
+
(
)
a
2
x t n
2
+
(
) ...
hq b
1
f x t m
1
+
(
)
(
)
b
2
f x t m
2
+
(
)
(
) ...
+
+
[
]
+
+
+
=
x t h
+
(
)
x t h
-
(
)
2
hf x t
( )
(
)
+
=
E
13
h
3
f
'
( )
=
t
true path
t
t+2h
FIGURE 21. Graph of the Mid-Point method
x
x(t)
t+h
t+3h
t-h
f(x(t))
f(x(t+h))
f(x(t+2h)
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