Advanced integration methods

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 $x_n$ to $x_{n+1}$ without needing to know the solutions at $x_{n-1}$, $x_{n-2}$, etc. There is a broad class of more sophisticated integration methods, known as multi-step methods, which utilize the previously calculated solutions at $x_{n-1}$, $x_{n-2}$, etc. in order to evolve the solution from $x_n$ to $x_{n+1}$. Examples of these methods are the various Adams methods¹⁷ 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.