Fractional-Step Runge--Kutta Methods: Representation and Linear Stability Analysis

TL;DR

This paper presents a framework for representing and analyzing fractional-step Runge--Kutta methods using the GARK approach, clarifying their stability and structural properties.

Contribution

It introduces a unified representation linking fractional-step methods to GARK methods, enabling stability analysis and insights into method design choices.

Findings

Relation between Butcher tableau and splitting coefficients

Impact of sub-integrator order and application sequence on stability

Role of negative splitting coefficients in stability

Abstract

Fractional-step methods are a popular and powerful divide-and-conquer approach for the numerical solution of differential equations. When the integrators of the fractional steps are Runge--Kutta methods, such methods can be written as generalized additive Runge--Kutta (GARK) methods, and thus the representation and analysis of such methods can be done through the GARK framework. We show how the general Butcher tableau representation and linear stability of such methods are related to the coefficients of the splitting method, the individual sub-integrators, and the order in which they are applied. We use this framework to explain some observations in the literature about fractional-step methods such as the choice of sub-integrators, the order in which they are applied, and the role played by negative splitting coefficients in the stability of the method.