Runge-Kutta-Fehlberg Method

The Runge-Kutta-Fehlberg method uses the analysis above with a Runge-Kutta method of order four and a Runge-Kutta method of order 5. Both methods are chosen such that only few extra calculations are needed to calculate , once has been obtained. When we introduce the definitions

;

;

;

;

;

;

Then the estimate is obtained using the Runge-Kutta method of order four, as

;

The estimate is obtained using the Runge-Kutta method of order five, as

;

Notice that the same intermediate results can be used in both methods and only 6 evaluations of are needed at every step.

In the Runge-Kutta-Fehlberg method, an initial value of is used to calculate and . Then a value of is calculated conservatively as

This is used to either

reject, if necessary, the initial choice of and repeat the calculations with ,

or to predict an appropriate step-size for the next step.

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The next worksheet will look at the application of this method and its accuracy.

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