Strong stability preserving explicit Runge-Kutta methods of maximal effective order

TL;DR

This paper explores the design of explicit SSP Runge-Kutta methods with maximal effective order, showing that higher effective order can be achieved with four stages, but classical order five SSP methods are impossible due to weight constraints.

Contribution

The paper introduces the concept of effective order to SSP methods, constructs optimal explicit SSP Runge-Kutta methods up to effective order four, and proves the limitations for order five.

Findings

Four-stage SSP methods with effective order four are possible.

Effective order five SSP methods require non-positive weights.

Numerical experiments confirm the practical effectiveness of the constructed methods.

Abstract

We apply the concept of effective order to strong stability preserving (SSP) explicit Runge-Kutta methods. Relative to classical Runge-Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods - like classical order five methods - require the use of non-positive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge-Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice.