Pseudo-Energy-Preserving Explicit Runge-Kutta Methods

This repository contains information and code to reproduce the results presented in the article

@article{barrios2025pseudo,
  title={Pseudo-Energy-Preserving Explicit {R}unge-{K}utta Methods},
  author={Barrios de Leon, Gabriel and Ketcheson, David I and Ranocha, Hendrik},
  journal={ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)},
  volume={59},
  number={2},
  pages={729--748},
  year={2025},
  month={03},
  eprint={2407.15365},
  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

@misc{barrios2024pseudoRepro,
  title={Reproducibility repository for
         "{P}seudo-Energy-Preserving Explicit {R}unge-{K}utta Methods"},
  author={Barrios de Leon, Gabriel and Ketcheson, David I and Ranocha, Hendrik},
  year={2024},
  howpublished={\url{https://github.com/Sondar74/PEP_Reproducibility2024}},
  doi={10.5281/zenodo.12699752}
}

Abstract

Using a recent characterization of energy-preserving B-series, we derive the explicit conditions on the coefficients of a Runge-Kutta method that ensure energy preservation (for Hamiltonian systems) up to a given order in the step size, which we refer to as the pseudo-energy-preserving (PEP) order. We study explicit Runge-Kutta methods with PEP order higher than their classical order. We provide examples of such methods up to PEP order six, and test them on Hamiltonian ODE and PDE systems. We find that these methods behave similarly to exactly energy-conservative methods over moderate time intervals and exhibit significantly smaller errors, relative to other Runge-Kutta methods of the same order, for moderately long-time simulations.

Numerical experiments

To reproduce the numerical experiments presented in this article, you need to install Python and Julia. The numerical experiments presented in this article were performed using Python v3.10.9 and Julia v1.10.4.

First, you need to download this repository, e.g., by cloning it with git or by downloading an archive via the GitHub interface.

To generate all figures of the paper, you can execute the Python code in the Jupyter notebook PEP_paper_Reproducibility.ipynb.

The Jupyter notebook loads some output data generated by the Julia code in the code_julia directory. To generate the data, 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

• Gabriel Barrios de Léon (Universidad de San Carlos de Guatemala)
• David I. Ketcheson (King Abdullah University of Science and Technology)
• 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!