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

2022/05/25 by Abner J. Salgado, Salgado, Abner J., Ignacio Tomaš +1
Computer Science · Mathematics · #34D20 #35K15 #35K20 #35K90 #35L10 #35L15 #35L20 #47J35 #58D25 #65L05 #65L06 #65M12 #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2205.13032

openalex publication_date 2022/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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