I was reading some work by Butcher and I came across Pade approximations and the correlation between them and stability functions for some Implicit Runge-Kutta methods. For example, in this Pade table for the exponential function, we see that the $(2,1)$ Pade approximation
corresponds to the stability function for the RADAU IIA method. Similarly, the $(2,2)$ Pade approximation
corresponds to the stability function of the 2-stage Gauss Runge-Kutta method. However, I'm not sure if other Pade approximations correspond to stability functions; For example, I can't find the numerical method whos stability function corresponds to the $(2,0)$ Pade approximation
So my question is two fold (I apologise for asking two questions in one post, it seems like a waste to create two new posts though):
- Is it always possible to generate an Implicit Runge-Kutta scheme whos stability function corresponds to a particular Pade approximation? And if so
- How do we construct such an implicit scheme?