A stability order barrier and SSP bound for Runge--Kutta methods with non-negative coefficients: AI-assisted proofs
Abstract
We prove three results about Runge--Kutta methods with non-negative coefficients. First, every s-stage implicit Runge--Kutta method of order at least two has SSP coefficient at most 2s; the bound is sharp and is attained by a known diagonally implicit method, so full implicitness gives no advantage over DIRK at second order. This extends the second-order DIRK result of Kocsis \& Németh to full coefficient matrices and settles a conjecture of Ketcheson, Macdonald & Gottlieb. These proofs were found with the assistance of AI tools, which uncovered an interesting relationship between the Butcher matrices for SSP methods and phase-type distributions, and led to the proof of the second result: a bound on the SSP coefficient of s-stage implicit Runge--Kutta method of order at least three. The third result in this work is a proof that a third-order Runge--Kutta method with entrywise non-negative A and b cannot be algebraically stable. This algebraically intensive proof was also AI-assisted. This paper describes the iterative process of AI-assisted results, and presents an approach that centers human understanding and creativity in the AI-assisted mathematical endeavor.
Related
- Stability in the determination of the coefficients for coupled relativistic Schrödinger system
- Optimization-based numerical methods for the two-and three-dimensional time harmonic Maxwell problem with sign-changing coefficients
- Well-posedness and Quantitative Stability for Degenerate Fokker--Planck Equations with Rough Coefficients
- Uniqueness and stability of nonlinear filtering equations with unbounded random coefficients
- Stability of the Lp Poisson-Neumann problem under small perturbations of the coefficients and the domain
BibTeX
@misc{gottlieb2026stability,
title = {{A stability order barrier and SSP bound for Runge--Kutta methods with non-negative coefficients: AI-assisted proofs}},
author = {Sigal Gottlieb and Zachary Grant and Lior Horesh and Lajos Lóczi and Avraham Shinnar and Barry Trager},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07981},
eprint = {2611.07981},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07981},
}
RIS
TY - UNPB
TI - A stability order barrier and SSP bound for Runge--Kutta methods with non-negative coefficients: AI-assisted proofs
AU - Sigal Gottlieb
AU - Zachary Grant
AU - Lior Horesh
AU - Lajos Lóczi
AU - Avraham Shinnar
AU - Barry Trager
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07981
UR - https://arxiv.org/abs/2611.07981
AB - We prove three results about Runge--Kutta methods with non-negative coefficients. First, every s-stage implicit Runge--Kutta method of order at least two has SSP coefficient at most 2s; the bound is sharp and is attained by a known diagonally implicit method, so full implicitness gives no advantage over DIRK at second order. This extends the second-order DIRK result of Kocsis \& Németh to full coefficient matrices and settles a conjecture of Ketcheson, Macdonald & Gottlieb. These proofs were found with the assistance of AI tools, which uncovered an interesting relationship between the Butcher matrices for SSP methods and phase-type distributions, and led to the proof of the second result: a bound on the SSP coefficient of s-stage implicit Runge--Kutta method of order at least three. The third result in this work is a proof that a third-order Runge--Kutta method with entrywise non-negative A and b cannot be algebraically stable. This algebraically intensive proof was also AI-assisted. This paper describes the iterative process of AI-assisted results, and presents an approach that centers human understanding and creativity in the AI-assisted mathematical endeavor.
ER -