Types of N-body ODE Integration Methods
June 9, 1996
At first glance, it might appear that these formula's would suffer the same problem
that adding more terms to the derivative of the Lagrange polynomial suffered from,
namely, the results would get worse with additional terms. After all, how much value is
knowing the position from a long time ago really going to help? Some of the results in
Marciniak's book seemed to confirm this suspicion (see page 28), but it turns out that the
formulas do not get worse with additional terms. There does, however, get to be a point
where the rounding error starts to get worse, so for practical reasons, the Adam-Bashford
methods are normally limited to around the 7th order.
In fact, the 7th order Adam-Bashford formula is the default for XStar because it is
the most efficient N-body ODE integration method for the default level of accuracy. The
high quality results that XStar gets from the Adam-Bashford method is somewhat surpris-
ing considering how little coverage most numerical analysis books give it. Several don't
cover it at all, others cover it only as part of predictor-corrector methods (discussed next).
The results shown in Marciniak's book do not look very promising either. Still, for XStar,
both the 4th order and the 7th order methods consistently ranked very high in terms of
being the "best" method as defined in Section 1.4 on page 20.
Sources: (1:343-6,12:61,4:373-6,13:315,15:210-1)
2.3 Predictor-Corrector Methods
Predictor-Corrector methods use the idea that once you have an approximate value
at a given time, there might be ways of improving this estimate. Predictor-Corrector
methods seem to have a lot of folklore associated with them. It is possible to call the cor-
rector method several times, but folklore has it that this is usually not worth while. Also, it
is possible to have a predictor and a corrector with different orders, but the folklore says
that the corrector should be equal to or only one order higher than the predictor. It is also
said that predictor and corrector formulas usually have "the same form", although it is not
clear exactly what is meant by that.
According to most of the books, Predictor-Corrector formulas are now passe. Now
a days people use either Runge-Kutta's or Gragg-Bulirsch-Stoer's methods. On the other
hand, the books imply that the straight multistep formulas were made obsolete by predic-
tor-corrector methods, and I didn't find this to be true.
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