arXiv subjects

Zachary Grant

Publications and source records attributed to Zachary Grant.

3 recordsLinked to original sources

arXiv2026

A stability order barrier and SSP bound for Runge--Kutta methods with non-negative coefficients: AI-assisted proofs

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.

math.NA↗

arXiv2018

A Strong Stability Preserving Analysis for Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions

High order strong stability preserving (SSP) time discretizations are often needed to ensure the nonlinear (and sometimes non-inner-product) strong stability properties of spatial discretizations specially designed for the solution of hyperbolic PDEs. Multiderivative time-stepping methods have recently been increasingly used for evolving hyperbolic PDEs, and the strong stability properties of these methods are of interest. In our prior work we explored time discretizations that preserve the strong stability properties of spatial discretizations coupled with forward Euler and a second derivative formulation. However, many spatial discretizations do not satisfy strong stability properties when coupled with this second derivative formulation, but rather with a more natural Taylor series formulation. In this work we demonstrate sufficient conditions for an explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and Taylor series formulation. We call these strong stability preserving Taylor series (SSP-TS) methods. We also prove that the maximal order of SSP-TS methods is p = 6, and define an optimization procedure that allows us to find such SSP methods. Several types of these methods are presented and their efficiency compared. Finally, these methods are tested on several PDEs to demonstrate the benefit of SSP-TS methods, the need for the SSP property, and the sharpness of the SSP time-step in many cases.

math.NA↗

arXiv2014

Strong Stability Preserving Multistep Runge-Kutta Methods

High-order spatial discretizations with strong stability properties (such as monotonicity) are desirable for the solution of hyperbolic PDEs. Methods may be compared in terms of the strong stability preserving (SSP) time-step. We prove an upper bound on the SSP coefficient of explicit multistep Runge--Kutta methods of order two and above. Order conditions and monotonicity conditions for such methods are worked out in terms of the method coefficients. Numerical optimization is used to find optimized explicit methods of up to five steps, eight stages, and tenth order. These methods are tested on the advection and Buckley-Leverett equations, and the results for the observed total variation diminishing and positivity preserving time-step are presented.

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