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

@article{ranocha2024multiderivative,
  title={Multiderivative time integration methods preserving nonlinear
         functionals via relaxation},
  author={Ranocha, Hendrik and Sch{\"u}tz, Jochen},
  journal={Communications in Applied Mathematics and Computational Science},
  year={2024},
  month={06},
  volume={19},
  issue={1},
  doi={10.2140/camcos.2024.19.27},
  eprint={2311.03883},
  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{ranocha2023multiderivativeRepro,
  title={Reproducibility repository for
         "{M}ultiderivative time integration methods preserving nonlinear
         functionals via relaxation"},
  author={Ranocha, Hendrik and Sch{\"u}tz, Jochen},
  year={2023},
  howpublished={\url{https://github.com/ranocha/2023_multiderivative_relaxation}},
  doi={10.5281/zenodo.10057727}
}

We combine the recent relaxation approach with multiderivative Runge-Kutta methods to preserve conservation or dissipation of entropy functionals for ordinary and partial differential equations. Relaxation methods are minor modifications of explicit and implicit schemes, requiring only the solution of a single scalar equation per time step in addition to the baseline scheme. We demonstrate the robustness of the resulting methods for a range of test problems including the 3D compressible Euler equations. In particular, we point out improved error growth rates for certain entropy-conservative problems including nonlinear dispersive wave equations.

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 v1.9.3.

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. Other numerical experiments use MATLAB, based on source code contained in the directory code_matlab.

• Hendrik Ranocha (Johannes Gutenberg University Mainz, Germany)
• Jochen Schütz (Hasselt University, Belgium)

The code in this repository is published under the MIT license, see the LICENSE file. Some parts of the implementation are inspired by corresponding code of OrdinaryDiffEq.jl published also under the MIT license, see their license file.

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