Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators

TL;DR

This paper establishes the energy stability of the Active Flux method by formulating it within the framework of summation-by-parts operators, providing a new theoretical foundation for its analysis.

Contribution

It introduces a novel formulation of the Active Flux method using degenerate SBP operators to prove its energy stability.

Findings

Active Flux can be formulated with degenerate SBP operators.

The formulation provides a new approach to stability analysis.

The method is shown to be energy stable within this framework.

Abstract

The Active Flux method is a numerical method for conservation laws using a combination of cell averages and point values, based on ideas from finite volumes and finite differences. This unusual mix has been shown to work well in many situations. We expand the theoretical justifications of the Active Flux method by analyzing it from the point of view of summation-by-parts (SBP) operators, which are routinely used to analyze finite difference, finite volume, and finite element schemes. We show that the Active Flux method can be formulated using degenerate SBP operators, yielding a first and novel approach for showing the energy stability of the Active Flux method.