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.

翻译：我们研究对等隐含的龙格-库塔(DIRK)计划(DIRK)计划,将它应用于与Gelfand三重框架相适应的抽象演变问题;我们引入与这一环境相适应的新的稳定概念,并提供简单、必要和充分的条件,以核实DIRK计划在我们的意义上和在Bochner型规范中是否稳定;我们使用一些流行的DIRK计划,以说明满足所需结构稳定性特性的案例和不相容的案例;此外,在一些温和的结构条件下,我们可以保证在时间分解问题上离散的离散解决办法对家庭的影响。