Algebraic Conditions for Stability in Runge-Kutta Methods and Their Certification via Semidefinite Programming

TL;DR

This paper develops algebraic and computational methods to certify the stability of Runge-Kutta methods using semidefinite programming, enhancing the ability to verify stability properties rigorously.

Contribution

It introduces two approaches—sum-of-squares programming and sharpened algebraic conditions—for certifying $A$- and $A(eta)$-stability in Runge-Kutta methods, integrating theoretical and computational techniques.

Findings

Successfully certifies stability of several diagonally implicit schemes.

Provides a practical computational framework for stability certification.

Enhances algebraic conditions to incorporate Runge-Kutta order conditions.

Abstract

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.