Analysis of the ODE Integration Methods

June 9, 1996

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The predictor-corrector methods seemed to be slightly worse than the Adam-Bash-

ford method. The fact that

*f()*

has to be evaluated twice per step, and thus the predictor-

corrector methods need twice the step size, seems to cancels out the improved error term.

Apparently it is more important to have a smaller step size and thus make the function

appear smoother than it is to have a more accurate, but larger step.

The Runge-Kutta method is hampered because it has to take trial steps into the

future, but the movement of the stars changes how the gravitational force field will look in

the future. So, the rk4 method has to not only calculate

*f()*

four times, it also has to do four

trial star movements, which are not going to be completely accurate and is more expensive

for the N-body problem than many other ODEs.

The taylor3 method that I developed is good at low accuracy levels because its

error term has a very small constant. It also, by it's very structure, tends to cancel out the

effects of the overstep phenomenon which is one of the worst forms of errors. It not just a

coincidence that the taylor3 method is right at the borderline of being the most accurate

method. It was the method that I used to develop most of XStar with and so, if I had

noticed an excess amount of accuracy, I would have either increased the display rate or

increased the number of stars. So, all the other methods are having to compete with the

taylor3 method at the very peak of its efficiency.

This document is best viewed as n-body.pdf because the translation to html was buggy.