AI 中文总结
本文建立两级和三级DIRK格式的U-稳定性与B-稳定性等价，为B-稳定方法提供Bochner范数先验估计，还证明U-稳定DIRK格式的收敛性并推导梯度流的离散能量耗散不等式。
AI 中文摘要
在Abner J. Salgado与Ignacio Tomas的《对角隐式Runge-Kutta格式：离散能量平衡律与紧性性质》（发表于《J. Number. Math.》2023年第31卷第4期，第313-341页）中，提出了对角隐式Runge-Kutta（DIRK）格式的U-稳定性概念。本文建立了两级和三级DIRK格式的U-稳定性与B-（代数）稳定性之间的等价关系，进而为B-稳定方法提供了合适的Bochner范数先验估计。作为应用，首先通过能量估计和紧性，证明了U-稳定DIRK格式在极小正则性条件下对线性强制演化问题的收敛性；其次，将这些离散化方法应用于梯度流，由此推导离散局部能量耗散不等式，并提供反例以说明级能量单调性的局限性。
英文摘要
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.
Comments18 pages
