In this work, we present approaches to rigorously certify $A$- and $A(\alpha)$-stability in Runge-Kutta methods through the solution of convex feasibility problems defined by linear matrix inequalities. We adopt two approaches. The first is based on sum-of-squares programming applied to the Runge-Kutta $E$-polynomial and is applicable to both $A$- and $A(\alpha)$-stability. In the second, we sharpen the algebraic conditions for $A$-stability of Cooper, Scherer, T{\"u}rke, and Wendler to incorporate the Runge-Kutta order conditions. We demonstrate how the theoretical improvement enables the practical use of these conditions for certification of $A$-stability within a computational framework. We then use both approaches to obtain rigorous certificates of stability for several diagonally implicit schemes devised in the literature.

翻译：本文提出了通过求解线性矩阵不等式定义的凸可行性问题，来严格验证龙格-库塔方法中$A$-稳定性和$A(\alpha)$-稳定性的方法。我们采用两种途径：第一种基于应用于龙格-库塔$E$-多项式的平方和规划，适用于$A$-稳定性与$A(\alpha)$-稳定性验证；第二种则通过整合龙格-库塔阶条件，对Cooper、Scherer、Türke和Wendler提出的$A$-稳定性代数条件进行了强化。我们证明了该理论改进如何使这些条件能够在计算框架中实际应用于$A$-稳定性的严格验证。最后，我们运用这两种方法对文献中提出的若干对角隐式格式获得了严格的稳定性证明。