Types of N-body ODE Integration Methods

June 9, 1996

35

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)

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.

(14:589-592,4:380)

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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