Optimal monotonicity-preserving perturbations of a given Runge-Kutta method

TL;DR

This paper investigates how to optimally perturb Runge-Kutta methods to maximize their monotonicity-preserving step size, providing theoretical bounds and algorithms for both linear and nonlinear problems.

Contribution

It introduces a systematic approach to perturb Runge-Kutta methods to enhance their absolute monotonicity radius, including proofs, bounds, and optimal perturbation algorithms.

Findings

Perturbations can increase the radius of absolute monotonicity.

Upper bounds on the monotonicity radius are established.

Algorithms for computing optimal perturbations are developed.

Abstract

Perturbed Runge--Kutta methods (also referred to as downwind Runge--Kutta methods) can guarantee monotonicity preservation under larger step sizes relative to their traditional Runge--Kutta counterparts. In this paper we study, the question of how to optimally perturb a given method in order to increase the radius of absolute monotonicity (a.m.). We prove that for methods with zero radius of a.m., it is always possible to give a perturbation with positive radius. We first study methods for linear problems and then methods for nonlinear problems. In each case, we prove upper bounds on the radius of a.m., and provide algorithms to compute optimal perturbations. We also provide optimal perturbations for many known methods.