Abstract :
This paper concerns A-stability barriers for polynomial extrapolations of implicit Runge-Kutta methods. It is shown that, under some mild restrictions, a method of even order cannot admit an A-stable first extrapolation, while an odd order method cannot admit first and second extrapolations that are both A-stable. It is also shown that a method which is not A-stable cannot admit an A-stable first extrapolation, and furthermore, that neither the first nor the second extrapolation of a symmetric method can be A-stable. More specifically, the first and second extrapolations of the diagonal and second sub-diagonal of the Padé table for the exponential function are shown not to be A-stable as is the second extrapolation of the first sub-diagonal of the Padé table. The implications of these results for some well known classes of high order methods are discussed. Composite methods for which these barriers do not apply are studied, especially those formed by compositions of a symmetric method with an L-stable method. Those based on well known symmetric methods, in particular the 2-stage Gauss or 3-stage Lobatto IIIA, are shown to admit first and second but no higher L-stable extrapolations. Those based on methods whose stability functions are not Padé approximations are shown not to admit an A-stable first extrapolation. To date no third extrapolation of any method is known to be A-stable.
