Types of N-body ODE Integration Methods

June 9, 1996

38

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

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