 Types of N-body ODE Integration Methods
June 9, 1996
36
2.3.1 Modified Euler's Method (not implemented)
Euler's method assumes that the acceleration at time
t
is a good approximation for
the acceleration for the entire interval from
t
to
t+h
. A better approximation would be the
average of the acceleration at time
t
and at time
t+h
. This can be done by using Euler's
method to predict the value of
x(t+h)
and then correcting it as follows:
This method improves Euler's method from a first order to a second order formula
at the cost of one additional evaluation of
f()
. While this is an improvement, it is still too
low of an order to be useful.
Sources: (13:70,15:207,19:417-8)
formula uses the future value
, instead of just previous values of
f()
. To get this
initial estimate of
, we use the Adam-Bashford formula as a predictor, and then
use the Adam-Moulton formula to correct it. So the method has this form:
Where
r, s, a
1
, ... , a
n
, b
1
, ..., b
n
,
are constants
Sources: (14:590,12:62-3,4:382-4,1:346-8)
y
x t
( )
hf x t
( )
(
)
+
=
x t h
+
(
)
x t
( )
h f y
( )
f x t
( )
(
)
+
2
(
)
+
=
E
O h
3
( )
=
t
true path
t
t+h
FIGURE 20. Graph of the Modified Euler's method
corrected location
x
x(t)
f(x(t))
f(y)
predicted location
f x t h
+
(
)
(
)
f x t h
+
(
)
(
)
y
x t
( )
hr a
1
f x t
( )
(
)
a
2
f x t h
-
(
)
(
)
-
a
3
f x t
2
h
-
(
)
(
) ...
-
+
[
]
+
=
x t h
+
(
)
x t
( )
hs b
1
f y
( )
b
2
f x t
( )
(
)
b
3
f x t h
-
(
)
(
)
-
...
+
+
[
]
+
=