Embedded Methods for Error Estimation

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 

Example 4.2.3 One method of low order in that class is the RKF2(3) method 

Note that unlike the multistep case, for Runge-Kutta methods an error estimation always requires two methods of different order. 

Though it is always the error of the lower order method which is estimated, one often uses the higher order and more accurate method for continuing the integration process. This foregoing is called local extrapolation. This is also reflected in naming the method, i.e. 

This method uses six stages for 5 order result and one more for obtaining the 4 order result One clearly sees that the method saves one function evaluation by meeting the requirement (4-2.5) for the 5th order method which is used for local extrapolation. 

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