We study diagonally implicit Runge-Kutta (DIRK) schemes when applied to abstract evolution problems that fit into the Gelfand-triple framework. We introduce novel stability notions that are well-suited to this setting and provide simple, necessary and sufficient, conditions to verify that a DIRK scheme is stable in our sense and in Bochner-type norms. We use several popular DIRK schemes in order to illustrate cases that satisfy the required structural stability properties and cases that do not. In addition, under some mild structural conditions on the problem we can guarantee compactness of families of discrete solutions with respect to time discretization.

中文翻译：

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对角线隐式 Runge-Kutta 方案：离散能量平衡定律和紧性

当应用于适合 Gelfand-triple 框架的抽象进化问题时，我们研究了对角线隐式龙格库塔 (DIRK) 方案。我们引入了非常适合此设置的新颖稳定性概念，并提供简单、必要和充分的条件来验证 DIRK 方案在我们的意义上和 Bochner 型范数中是否稳定。我们使用几种流行的 DIRK 方案来说明满足所需结构稳定性属性的情况和不满足要求的情况。此外，在问题的一些温和结构条件下，我们可以保证离散解族在时间离散化方面的紧致性。