Order and Step Size Selection

We saw that the error in the numerical solution depends on the problem / and on the step size h. The step size influences both the dynamic behavior of the error equation and the size of the global error increment (4.1.15). 

In order to control the global error a step size is selected in such a way that the global error increment is approximately of the size of a given tolerance TOL. Meeting such a tolerance bound without overshooting it or being over accurate is the aim of a step size control mechanism. 

This example also shows the problems, when running a simulation with a fixed step size code. (Considering an obstacle on the road this example also demonstrates the problems when integrating a non smooth problem, see Ch. 6). 

For estimating the global error increment at step n, this step is made by two different methods, started with the same data. If we assume that this data is taken to be points of a solution trajectory, a comparison of these two numerical results can give an estimate of the global error increment. 

Let the method denoted by the superscript (1) be the explicit predictor method, then 4 = I- We obtain by using the matrix relation A B = BA)  

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