Initial Value Problems

In the previous chapter we derived a simple finite difference method, namely the explicit Euler method, and we indicated how this can be analysed so that we can make statements concerning its stability and order of accuracy. If Euler’s method is used with constant time step h then it is convergent with an error of order O(h) for all sufficiently smooth problems. Thus, if we integrate from 0 to 1 with step \(h = 1{0}^{-5}\), we will need to perform 105 function evaluations to complete the integration and obtain a solution with error O(h). To achieve extra accuracy using this method we could reduce the step size h. This is not in general efficient and in many cases it is preferable to use higher order methods rather than decreasing the step size with a lower order method to obtain higher accuracy. One of the main differences between Euler’s and higher order methods is that, whereas Euler’s method uses only information involving the value of y and its derivative (slope) at the start of the integration interval to advance to the next integration step, higher order methods use information at more than one point. There exist two important classes of higher order methods that we will describe here, namely Runge-Kutta methods and linear multistep methods.