ABSTRACT
The first types of numerical methods for ordinary differential equations are classified as single-step processes. The best-known single-step methods are those based on Taylor series and the Runge-Kutta formulae. The idea of step-by-step computation, and that of order reduction, is fundamental to the numerical analysis of differential equations. The importance of Taylor series expansions goes well beyond their direct application as numerical methods, as will become clear as the subject is developed. The application of numerical methods to problems which can be solved by analytical means is a sensible way of gaining experience of these techniques and also of building confidence in them. In order to evaluate any new computational scheme one must ascertain its behaviour when it is applied to problems with known solutions. A comparison of the numerical solution with the true solution of the original problem yields an overall or global error.
