# Initial-value problems: multi-step methods

## Abstract

The one-step finite-difference methods of the previous chapter, with the possible exception of the trapezoidal rule method, are basically rather too uneconomic for general use. Their local truncation error is of rather low order, with the result that to achieve good accuracy we need to use either: (a) a rather small step-by-step interval, involving many steps to cover a specified range; or (b) one of our correcting devices which involve some additional computation and more computer programming. The special one-step methods of the previous chapter, the Taylor series method and the explicit Runge-Kutta methods, do not share these disadvantages, but they involve extra computer storage, the Taylor series method involves a possibly large amount of non-automatic differentiation, and with the Runge—Kutta methods we have to compute possibly rather complicated expressions several times in each step. Moreover, both these methods have rather poor partial stability properties.

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