This paper deals with stability of classical Runge-Kutta collocation methods. When such methods are embedded in linearly implicit methods as developed in [12] and used in [13] for the time integration of nonlinear evolution PDEs, the stability of these methods has to be adapted to this context. For this reason, we develop in this paper several notions of stability, that we analyze. We provide sufficient conditions that can be checked algorithmically to ensure that these stability notions are fulfilled by a given Runge-Kutta collocation method. We also introduce examples and counterexamples used in [13] to highlight the necessity of these stability conditions in this context.

翻译：本文研究经典龙格-库塔配置方法的稳定性。当此类方法嵌入文献[12]中发展的线性隐式方法，并用于文献[13]中对非线性发展偏微分方程进行时间积分时，其稳定性需要适应这一应用场景。为此，本文提出并分析了若干稳定性概念。我们给出了可通过算法验证的充分条件，以确保给定龙格-库塔配置方法满足这些稳定性要求。此外，本文还介绍了文献[13]中使用的示例和反例，以凸显在此背景下这些稳定性条件的必要性。