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

Austin Juhl, David Shirokoff

TL;DR

The paper develops convex optimization-based certificates for $A$- and $A(\alpha)$-stability of Runge–Kutta methods by solving LMIs, using two complementary routes: SOS testing of the $E$-polynomial and CSTW algebraic conditions. It strengthens the CSTW criteria by integrating RK order conditions (main theorem CSTWModified), enabling rigorous computer-assisted stability certification. The authors demonstrate the framework on diagonally implicit RK schemes, obtaining fully rigorous certificates and, in several cases, explicit angle bounds for $A(\alpha)$-stability. The work provides a practical, software-friendly approach to verify stability properties of modern RK schemes with potential applications in industry and software verification.

Abstract

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, T{ü}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.

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

TL;DR

The paper develops convex optimization-based certificates for - and -stability of Runge–Kutta methods by solving LMIs, using two complementary routes: SOS testing of the -polynomial and CSTW algebraic conditions. It strengthens the CSTW criteria by integrating RK order conditions (main theorem CSTWModified), enabling rigorous computer-assisted stability certification. The authors demonstrate the framework on diagonally implicit RK schemes, obtaining fully rigorous certificates and, in several cases, explicit angle bounds for -stability. The work provides a practical, software-friendly approach to verify stability properties of modern RK schemes with potential applications in industry and software verification.

Abstract

In this work, we present approaches to rigorously certify - and -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 -polynomial and is applicable to both - and -stability. In the second, we sharpen the algebraic conditions for -stability of Cooper, Scherer, T{ü}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 -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.
Paper Structure (32 sections, 7 theorems, 89 equations, 1 figure)

This paper contains 32 sections, 7 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Contributions and Organization of the Paper
  4. Runge-Kutta Background
  5. Linear Stability
  6. Accuracy and Order Conditions
  7. Linear Matrix Inequalities Background
  8. Feasibility and Convexity of Linear Matrix Inequalities
  9. Non-negative Polynomials as Linear Matrix Inequalities
  10. $A$- and $A(\alpha)$-Stability as Linear Matrix Inequalities
  11. The $E$-polynomial LMI for $A(\alpha)$-stability
  12. The CSTW Algebraic Conditions and LMI for $A$-stability
  13. Main Lemma: Sharpening the CSTW Conditions
  14. Examples Highlighting Lemma \ref{['thm:nullvec']}
  15. Examples and Results for Runge-Kutta Schemes
  16. ...and 17 more sections

Key Result

Lemma 4.1

A scheme is $A(\alpha)$-stable if:

Figures (1)

  • Figure 1: The blue line is a visualization of $\mathcal{R}(\hbox{\boldmath$A$},\boldsymbol{b})$ for SDIRK(3,2) in Example \ref{['Example:DIRK3_2']}. Note the set $\mathcal{R}$ has dimension $1$, as characterized by Theorem \ref{['Thm:CSTWModified']}, which is lower than the expected $3$ dimensional set suggested by Theorem \ref{['SW:1994']}.

Theorems & Definitions (18)

  • Lemma 4.1
  • Theorem 4.2
  • Remark 4.3
  • Remark 4.4: Degenerate stability functions
  • Lemma 4.5
  • proof
  • Remark 4.6
  • Theorem 4.7: Main Result, CSTW Conditions with order conditions
  • proof
  • Example 4.8: SDIRK(3,2)
  • ...and 8 more