Dense output for strong stability preserving Runge-Kutta methods

TL;DR

This paper explores the development of dense output formulas for SSP Runge-Kutta methods, establishing their existence limits and providing explicit formulas for certain orders while proving higher-order formulas are impossible.

Contribution

It introduces a general approach for first-order SSP dense output and constructs second-order formulas for optimal SSP methods, also proving non-existence of higher-order SSP dense outputs.

Findings

First-order SSP dense output formulas are possible.

Second-order SSP dense output formulas are developed for specific methods.

SSP dense output formulas of order 3 or higher do not exist.

Abstract

We investigate dense output formulae (also known as continuous extensions) for strong stability preserving (SSP) Runge-Kutta methods. We require that the dense output formula also possess the SSP property, ideally under the same step-size restriction as the method itself. A general recipe for first-order SSP dense output formulae for SSP methods is given, and second-order dense output formulae for several optimal SSP methods are developed. It is shown that SSP dense output formulae of order 3 and higher do not exist, and that in any method possessing a second-order SSP dense output, the coefficient matrix A has a zero row.