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

This repository contains information and code to reproduce the results presented in the article bibtex @online{barsukow2025stability, title={Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators}, author={Barsukow, Wasilij and Klingenberg, Christian and Lechner, Lisa and Nordstr{\"o}m, Jan and Ortleb, Sigrun and Ranocha, Hendrik}, year={2025}, month={07}, eprint={2507.11068}, eprinttype={arxiv}, eprintclass={math.NA} }

If you find these results useful, please cite the article mentioned above. If you use the implementations provided here, please also cite this repository as bibtex @misc{barsukow2025stabilityRepro, title={Reproducibility repository for "{S}tability of the Active Flux Method in the Framework of Summation-by-Parts Operators"}, author={Barsukow, Wasilij and Klingenberg, Christian and Lechner, Lisa and Nordstr{\"o}m, Jan and Ortleb, Sigrun and Ranocha, Hendrik}, year={2025}, howpublished={\url{https://github.com/ranocha/2025_active_flux_sbp}}, doi={10.5281/zenodo.15861045} }

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.

Numerical experiments

To reproduce the numerical experiments presented in this article, you need to install Julia. The numerical experiments presented in this article were performed using Julia v1.10.9.

First, you need to download this repository, e.g., by cloning it with git or by downloading an archive via the GitHub interface. Then, you need to start Julia in the code_julia directory of this repository and follow the instructions described in the README.md file therein.

Authors

• Wasilij Barsukow
• Christian Klingenberg
• Lisa Lechner
• Jan Nordström
• Sigrun Ortleb
• Hendrik Ranocha (Johannes Gutenberg University Mainz, Germany)

License

The code in this repository is published under the MIT license, see the LICENSE file.

Disclaimer

Everything is provided as is and without warranty. Use at your own risk!