Stability of Implicit Runge-Kutta Methods

The main reason for using implicit Runge-Kutta methods is due the excellent stability properties of some of the methods in this class. Again, we consider stability for the linear test equation and obtain by applying (4.3.3) 

Solving this linear system for the stage values X and inserting these values into the last equation results in 

In Fig. 4.8 the stability region of the three stage Radau Ila method is displayed. One realizes that the stability of the Radau method is much alike the stability of the implicit Euler method, though the three stage Radau method has order 5. Again we note the property i (0) = 1 which corresponds to zero stability in the multistep case. 

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Stability of Implicit Runge-Kutta Methods for DAEs... 

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