In this chapter the discretisations which were briefly introduced in the first chapter will be discussed and analysed in more detail. First we consider the construction of one-step methods, in particular Runge-Kutta methods, in 1. Next we investigate the local discretisation error and the consistency (order) of a method in 2. In 3 the notion of convergence (of the numerical approximation to the exact solution) and a theorem, with. sufficient conditions to achieve this, are treated. Since the result of the latter theorem is rather crude, 4 is devoted to giving a more precise estimate of the magnitude of the global discretisation error (viz. by deriving the leading term in the asymptotic expansion). In 5 two methods are given to estimate the in practice more important local error. In 6 this idea is implemented in a practical recommendation for an adaptive integrator. An analysis of the actual errors is also given, for various tolerance criteria, and a number of examples illustrate its effectiveness.

1. Runge-Kutta Methods

One-step methods are not only mathematically quite natural (as we shall see in Chapter VII on multistep methods); they are also attractive because they do not complicate the computations when the step size is changed. An important class of one-step methods are the Runge-Kutta methods. Consider the equation (cf. (I.6.3))

By taking t= t _i and T= t _{i +1} we may approximate the integral in (1.1) by a quadrature formula, i.e. the discrete sum (cf. Appendix A)

where h:=