Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators

This repository contains some code used in the following article which is available on arXiv.

@article{ranocha2020discrete,
  title={Discrete Vector Calculus and {H}elmholtz {H}odge Decomposition
         for Classical Finite Difference Summation by Parts Operators},
  author={Ranocha, Hendrik and Ostaszewski, Katharina and Heinisch, Philip},
  journal={Communications on Applied Mathematics and Computation},
  volume={2},
  pages={581--611},
  year={2020},
  month={02},
  eprint={1908.08732},
  eprinttype={arxiv},
  eprintclass={math.NA},
  doi={10.1007/s42967-019-00057-2}
}

If you find these results useful, please cite the article mentioned above. If you use the implementations provided here, please cite this repository as

@misc{ranocha2019discreteRepro,
  title={{2019\_SBP\_vector\_calculus\_REPRO}.
         {D}iscrete Vector Calculus and {H}elmholtz {H}odge Decomposition for
         Classical Finite Difference Summation by Parts Operators},
  author={Ranocha, Hendrik and Ostaszewski, Katharina and Heinisch, Philip},
  year={2019},
  month={08},
  howpublished={\url{https://github.com/IANW-Projects/2019_SBP_vector_calculus_REPRO}},
  doi={10.5281/zenodo.3375170}
}

Brief Description

This repository contains two Jupyter notebooks containing Julia code.

• notebooks/grid_oscillations__used.ipynb Here, grid oscillations associated to nullspace consistent first derivative summation by parts (SBP) operators are visualised. The corresponding figures in the article can be reproduced using this code and interactive experiments are set up.
• notebooks/projections__used.ipynb Here, a discrete Helmholtz Hodge decomposition using classical finite difference SBP operators is conducted for several parameters. Besides visualisations of the decomposition in two and three space dimensions, convergence studies and applications to wave mode analysis of the magnetohydrodynamic (MHD) equations are presented. This notebook contains the code for all other numerical examples and figures in the article.

Disclaimer

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