  The use of high order fully implicit Runge-Kutta methods is of significant
importance in the context of the numerical solution of transient partial
differential equations, in particular when solving large scale problems due to
fine space resolution with many millions of spatial degrees of freedom and long
time intervals. In this study we consider strongly A-stable implicit
Runge-Kutta methods of arbitrary order of accuracy, based on Radau quadratures,
for which efficient preconditioners have been introduced. A refined spectral
analysis of the corresponding matrices and matrix-sequences is presented, both
in terms of localization and asymptotic global distribution of the eigenvalues.
Specific expressions of the eigenvectors are also obtained. The given study
fully agrees with the numerically observed spectral behavior and substantially
improves the theoretical studies done in this direction so far. Concluding
remarks and open problems end the current work, with specific attention to the
potential generalizations of the hereby suggested general approach.
