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.

翻译：在瞬时部分差异方程式的数字解决方案中,使用完全隐含的高顺序龙格-库塔方法非常重要,特别是在用数百万空间自由度和较长时间间隔解决由于细微的空间分辨率造成的大规模问题时。在本研究报告中,我们强烈地认为,基于Radau二次曲线的可容许的隐含龙格-库塔任意顺序的方法非常重要,已经为此引入了高效的先决条件。对相应的矩阵和矩阵序列进行了精细的光谱分析,既包括电子元值的本地化,也包括零星分布的全球分布。还获得了电子元体的具体表达。本研究报告完全同意数字观测到的光谱行为,并大大改进了迄今为止在这一方向上进行的理论研究。结束目前的工作,总结和公开问题,并特别关注本建议的一般方法的可能普遍性。