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.