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Of course, Runge-Kutta methods are not the last word in integrating o.d.e.s. Far from
it! Runge-Kutta methods are sometimes referred to as *single-step* methods,
since they evolve the solution from to without needing to
know the solutions at , , *etc.* There is a broad
class of more sophisticated integration methods, known as *multi-step* methods,
which utilize the previously calculated solutions at , ,
*etc.* in order to evolve the solution from to . Examples of these
methods are the various Adams methods^{17} and the various Predictor-Corrector
methods.^{18} The main advantages of
Runge-Kutta methods are that they are easy to implement, they are very stable,
and they are ``self-starting'' (*i.e.*, unlike muti-step methods, we do not
have to treat the first few steps taken by a single-step integration method as special cases).
The primary disadvantages of Runge-Kutta methods are that they require significantly
more computer time than multi-step methods of comparable accuracy, and
they do not easily yield good *global* estimates of the truncation error. However, for the straightforward
dynamical systems under investigation in this course, the advantage of the relative simplicity and
ease of use of Runge-Kutta methods far outweighs the disadvantage of their
relatively high computational cost.

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** Up:** Integration of ODEs
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Richard Fitzpatrick
2006-03-29