3.1. Introduction

One-step methods, like Runge-Kutta formulae, used for solving Ordinary Differential Equations (ODEs) require the solution value at one mesh point to compute the solution at the next. The methods thus obtained are simple and easy to program. On the other hand, they are inefficient in that they do not make use of all the previous computed results.

It seems plausible that more accuracy can be obtained if the values of \(\,y_{j+1}\,\) is made to depend not only on \(y_j\) but also on \(y_{j-1}\) and \(y_{j-2}\), for example. However, one-step methods are useful for calculating the first few starting points. Once the values at a number of points have been computed, they can be used in the computation of later points using multistep methods which are economical as well as being generally more accurate.

Methods using more than one value of the dependent variable to determine the approximation at the next mesh points are called multivalue or multistep methods. A method is called a \(k\)-step method if the values \(\,y_j, y_{j-1},...,y_{j+1-k}\,\) are required for the calculation of \(y_{j+1}\).

The most common method used to derive the difference-equations for multistep methods is by the polynomial interpolation. Another method for deriving the formulae for multistep methods is called the method of undetermined coefficients – (covered in the second year Numerical Methods unit).

The interpolating polynomials are used in many areas of numerical analysis. In this chapter, we shall review some properties of the interpolating polynomials and develop means for working with functions that are stored in tabular form.

Note

• One-step Methods: \((y_j) \Longrightarrow y_{j+1}\)

• Multi-step Methods: \((y_{j-k}, y_{j-k+1}, \ldots, y_j) \Longrightarrow y_{j+1}\)