Theory of the N-Body Problem
June 9, 1996
20
So, there is really some other, unknown, term that is on the order as the square of the step
size
h
times the value of the rate of change of the acceleration at some time in the time
period in which this move took place. Taylor's theorem says that
, but that
the exact value of can not be known.
Using Taylor's theorem to derive a more accurate approximation of the function
can be done by using additional terms. As an example, we could use:
The error in this formula, i.e. the difference between the infinite Taylor series and the
Taylor series that has been truncated
1
after 5 terms will be no larger than
.
(16:580-
2)
While the Taylor series could be used to create a very accurate solution, the problem
with it is that even the third derivative of
x(t)
can be complicated and expensive to calcu-
late in the case of the N-body problem.
All the other methods used to solve the N-body differential equation end up having
a similar sort of error term. That is, the error will have some constant times, some power
of the step size times, some complicated function evaluated during the time interval
(t ...
t+h)
. The constant is often hard to calculate, the function is very hard to calculate, the time
value that the function is evaluated at is impossible to calculate, so the only easily known
quantity of this error term is the power of the step size.
(16:580-2)
The error term from above
would normally be denoted as
O(h
5
)
, that is, it is on the order of
h
5
. Thus, the formula is
accurate to
O(h
4
)
and as a result, this method would be known as a "fourth order" approx-
imation of the exact solution.
1.4 What Is the "Best" Method?
Before an evaluation of the different methods of integrating ODEs can be made,
objective criteria must be defined. For most N-body problems, we can use this definition.
Definition:
One method is
better
than another method if, in a fixed amount of
time, it is able to calculate the positions and velocities with greater
accuracy.
1.4.1 The Efficiency of a Method
In the definition of a "better" method, the phrase "in a fixed amount of time" is
very important. It doesn't do any good to have a routine that can calculate the movements
of the stars more accurately if it also takes longer to do the calculations than another
method. The quicker method could be made more accurate by just decreasing the step size
a little. For the same reason, it is also meaningless to just compare "which is more accu-
rate at a given step size", some methods do much more work per step.
1. Discretization error is also known as truncation error for this reason.
t
...
t h
+
(
)
x t h
+
(
)
x t
( )
hx
'
t
( )
12
h
2
x
''
t
( )
1
2 3
h
3
x
'''
t
( )
1
4!
h
4
x
4
( )
t
( )
+
+
+
+
=
1
5!
h
5
x
5
( )
( )