Theory of the N-Body Problem
June 9, 1996
29
Marciniak's book
(12:72)
, his 7th order Adam-Bashford method gave much worse results
than his 4th order Adam-Bashford method in a particular case. In my testing I found a case
where the 4th order Adam-Bashford method with an accuracy parameter of one (i.e. the
command `xstar -m ab4 -a 1') gave much better results than the 7th order Adam-Bashford
method with the higher accuracy parameter of -a 4. It also gave better results than -m ab4
with the higher accuracy parameter -a 2. Obviously, sometimes a method just gets lucky
(or unlucky).
One more point should be made when estimating which method of integrating an
ODE might be better. Each time the derivative of
f()
is taken, a constant is pulled out due
to the inverse square property of
f().
So the first derivative of
f()
cause the constant -2 to be
pulled out, the second derivative pulls out the constant -3. The constant of the error term
of a 7th order method has to be multiplied by -9! = -362880. This is a very large constant
that the higher order method has to over come and one of the reasons why lower order
methods are more accurate when lower accuracy parameter settings are used.