Global error increment

Both, the global error increment and the local residual, have the same asymptotic behavior for h tending to zero but with a different magnitude for finite h. We will see later that for so-called stiff ODEs the difference in this constant can be of importance. 

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. 

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. 

Example 4.1.13 In order to illustrate the estimates of the global error increment the unconstrained truck is simulated running over a 12 cm high road hump described by the input function u t) = — and its derivative (see Sec. 1.3.1). 

The simulation is performed by using the explicit Euler method as predictor and the implicit Euler method as corrector (with fixed point iteration iterated %ntil convergence , see Sec. 4 1-1)- I 4- l he results for the rotation of the chassis obtained with h = 10 are plotted. The thickness of the band around this solution curve corresponds to the estimated global error increment magnified by 5 10. This figure reflects clearly the influence of the second derivative of the solution on the size of the error. 

Figure 4.5 Simulation result of the truck running over an obstacle together with the estimated global error increment (magnified by 5 10 ).

Variable step size codes adjust the step size in such a way, that the global error increment is kept below a certain tolerance threshold TOL. This requires a good estimation of this quantity. Like in the multistep case, the error can be estimated by comparing two different methods. Here, a Runge-Kutta method of order p and another method of order p -f 1 is taken to perform a step from tn to tn+i, say. The global error increment of the p order method is... 

---