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

TL;DR

This paper analyzes diagonally implicit Runge-Kutta schemes for abstract evolution problems, introducing new stability notions, verifying their conditions, and establishing compactness of discrete solutions under certain conditions.

Contribution

It introduces novel stability concepts for DIRK schemes in the Gelfand-triple setting and provides criteria to verify stability and compactness properties.

Findings

Certain DIRK schemes satisfy the new stability conditions.

Some schemes do not meet the stability criteria.

Under mild conditions, discrete solutions are compact in time.

Abstract

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.