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：在瞬态偏微分方程数值求解中，高阶全隐式龙格-库塔方法具有重要应用价值，特别是当因精细空间分辨率导致数百万空间自由度和长时程大规模问题求解时。本研究基于Radau求积公式，考虑任意精度阶数的强A-稳定隐式龙格-库塔方法，并引入其高效预条件子。本文对相应矩阵及矩阵序列进行精细谱分析，涵盖特征值定位与渐近全局分布两方面，同时获得特征向量的具体表达式。研究成果完全吻合数值观测到的谱特征，并显著改进了现有理论分析。最后通过结论与开放问题收尾，特别关注本文所提出一般化方法的潜在推广方向。