Analysis of the ODE Integration Methods
June 9, 1996
Our prediction is that, as extrapolation methods like
Bulirsch-Stoer continue to gain sophistication, they will
eventually beat out PC methods in all applications. We are
willing, however, to be corrected.(14:592)
Let's look at the properties of our particular problem. First, the evaluation of the
right-hand side, i.e.
, is
expensive, even if a fast force evaluation method is used.
The star system movement is normally very smooth, but it can also have singularities or
poles in the complex plane when stars pass close to each other. The ODE integration rou-
tines need to be accurate, but since the results are just used to draw pictures, very high
accuracy is not really required. All that is required is that it has to be accurate enough to
avoid the worst of the movement errors.
Another atypical property of our star movement problem is that the smaller the
step size is, the less a `near collision' looks like a singularity and the more it looks like a
smooth path. This is because the stars tend to move around each other so the point of the
singularity (i.e., the other star) tends to move out of the way. Thus, by using a simpler
method such as the Adam-Bashford method which requires a very smooth function, you
can make the step size smaller and get a smooth function. If a more expensive method,
such as the Gragg-Bulirsch-Stoer's is used, a much large step size must be used and thus
you get the singularities that it can't handle.
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