Bochner Stability for B-stable DIRK Schemes

Abstract

In Abner J. Salgado and Ignacio Tomas. Diagonally implicit Runge-Kutta schemes: discrete energy-balance laws and compactness properties. J. Number. Math., 31(4):313-341, 2023, the notion of $U$-stability for Diagonally Implicit Runge-Kutta (DIRK) schemes was introduced. Here we establish the equivalence between $U$- and $B$- (algebraic) stability for two- and three-stage DIRK schemes, which then} provides suitable Bochner norm a priori estimates for $B$-stable methods. As applications, we first prove the convergence of $U$-stable DIRK schemes for linear coercive evolution problems under minimal regularity via energy estimates and compactness. Second, we apply these discretizations to gradient flows, which allow us to derive discrete local energy dissipation inequalities and provide counterexamples that demonstrate the limitations of stagewise energy monotonicity.