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：在瞬态偏微分方程的数值求解中，高阶全隐式龙格-库塔方法具有重要意义，尤其当因精细空间分辨率（涉及数百万空间自由度）与长时间区间而需要求解大规模问题时。本研究基于拉道求积法，考虑任意精度阶数的强A-稳定隐式龙格-库塔方法，并引入其高效预处理子。针对相应矩阵与矩阵序列，本文从特征值的局域化与渐近全局分布两方面展开精细谱分析，同时获得了特征向量的具体表达式。该研究结果与数值观测的谱行为完全吻合，并显著改进了迄今为止在该方向上的理论研究工作。本文以结论性评述与开放性问题作结，特别关注所提出通用方法的潜在推广方向。