 Types of N-body ODE Integration Methods
June 9, 1996
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2.5 The Gragg-Bulirsch-Stoer Method (-m gpemce8)
The Gragg-Bulirsch-Stoer method is literally in a class by itself and seems to be
the current favorite for solving this type of differential equation.
The basic idea behind this method is that as the step size is decreased, the answer
should become more accurate. If a large interval
H
is taken and a variety of smaller step
sizes are used over that interval, then you will get a variety of different answers but the
answers should converge toward the correct outcome as the step size gets smaller. (See
FIGURE 22.) Well, why not take these converging answers and create a polynomial
approximation of the answers as a function of the step size? Then the magic step size of
h=0
can be put into this polynomial, and the result should be a very good approximation
of the answer as if no discretization had ever been done. (In FIGURE 23., the polynomial
is shown as a function of the number of steps, but to make it a function of the step size is
fairly straight forward.) This same basic idea is also used in Romberg's integration, and
the whole idea was pioneered by Richardson with his idea of "extrapolating to the limit."
t
x
true path
8 steps
4 steps
2 steps
t
t+H
FIGURE 22. Results of using smaller step sizes
1 step
Number
of steps
x
1 2
4
8
x value for an infinite number of steps
FIGURE 23. Polynomial approximation of x based on step size
polynomial approximation