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

@online{winters2025numerical,
  title={Numerical flux functions that give provable bounds for nonlinear initial boundary value problems with open boundaries},
  author={Winters, Andrew R and Kopriva, David A and Nordström, Jan},
  year={2025},
  month={11},
  eprint={2511.04197},
  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{winters2025numericalRepro,
  title={Reproducibility repository for
         "{N}umerical flux functions that give provable bounds for nonlinear initial boundary value problems with open boundaries"},
  author={Winters, Andrew R and Kopriva, David A and Nordström, Jan},
  year={2025},
  howpublished={\url{https://github.com/andrewwinters5000/2025_nonlinear_bndy_flux}},
  doi={https://doi.org/10.5281/zenodo.17533030}
}

We present a strategy for interpreting nonlinear, characteristic-type penalty terms as numerical boundary flux functions that provide provable bounds for solutions to nonlinear hyperbolic initial boundary value problems with open boundaries. This approach is enabled by recent work that found how to express the entropy flux as a quadratic form defined by a symmetric boundary matrix. The matrix formulation provides additional information for how to systematically design characteristic-based penalty terms for the weak enforcement of boundary conditions. A special decomposition of the boundary matrix is required to define an appropriate set of characteristic-type variables. The new boundary fluxes are directly compatible with high-order accurate split form discontinuous Galerkin spectral element and similar methods and guarantee that the solution is entropy stable and bounded solely by external data. We derive inflow-outflow boundary fluxes specifically for the Burgers equation and the two-dimensional shallow water equations, which are also energy stable. Numerical experiments demonstrate that the new nonlinear fluxes do not fail in situations where standard boundary treatments based on linear analysis do.

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 v1.11.3.

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 directory of this repository and follow the instructions described in the README.md file therein.

• Andrew R. Winters (Linköping University, Sweden)
• David A. Kopriva (Florida State University, Florida, USA and San Diego State University, California, USA)
• Jan Nordström (Linköping University, Sweden and University of Johannesburg, South Africa)

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

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