Stability of Runge-Kutta Methods

Similar to the discussion in the multistep case we investigate the stability of Runge-Kutta methods for finite step sizes h by considering the linear test equation x = x, cf. (4.1.19). 

Applying the Runge-Kutta method (4.2.2) to that equation results in 

By inserting the stage values into the final equation we get 

The function i (C — (P is called stability function of the Runge-Kutta method, cf. the definition of for multistep methods Sec. 4.1.4. 

A Runge-Kutta method is stable for a given step size h and a given complex parameter A iff / (/iA) 1. Then, an error in xq is not increased by applying (4.2.12). 

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Dekker, K., Verver, J.G. Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Eqnations. North-Holland, Amsterdam (1984)... 

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