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

Diagonally implicit Runge-Kutta schemes: Discrete energy-balance laws and compactness properties