We prove that Runge-Kutta (RK) methods for numerical integration of arbitrarily large systems of Ordinary Differential Equations are linearly stable. Standard stability arguments -- based on spectral analysis, resolvent condition or strong stability, fail to secure the stability of arbitrarily large RK systems. We explain the failure of different approaches, offer a new stability theory and demonstrate a few examples.

翻译：我们证明了用于任意大规模常微分方程组数值积分的龙格-库塔（RK）方法是线性稳定的。基于谱分析、预解条件或强稳定性的标准稳定性论证，无法确保任意大规模RK系统的稳定性。本文阐释了不同方法的失效原因，提出了一种新的稳定性理论，并给出若干实例。