Analysis of the ODE Integration Methods
June 9, 1996
54
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.
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