2024/05/22 by Austin Juhl, Juhl, Austin, David Shirokoff +1
Computer Science · Mathematics · #65L06 #65L07 #65L20 #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2405.13921
openalex publication_date 2024/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we present approaches to rigorously certify A- and A(α)-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(α)-stability. In the second, we sharpen the algebraic conditions for A-stability of Cooper, Scherer, Türke, 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.