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https://github.com/ranocha/convexrelaxationrungekutta

Relaxation Runge-Kutta Methods for Convex Functionals
https://github.com/ranocha/convexrelaxationrungekutta

jupyter-notebook runge-kutta-methods stability

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Relaxation Runge-Kutta Methods for Convex Functionals

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# Relaxation Runge-Kutta Methods for Convex Functionals



[![License: MIT](https://img.shields.io/badge/License-MIT-success.svg)](https://opensource.org/licenses/MIT)

[![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.3066518.svg)](https://doi.org/10.5281/zenodo.3066518)



Relaxation Runge-Kutta methods are modifications of Runge-Kutta methods that

enforce conservation, dissipation, or other solution properties with respect

to any convex functional by the addition of a *relaxation parameter* that

multiplies the Runge-Kutta update at each time step.

Moreover, other desirable stability (such as strong stability preservation)

and efficiency (such as low storage requirements) properties are preserved.

The technique can be applied to both explicit and implicit Runge-Kutta

methods and requires only a small modification to existing implementations.

The computational cost at each step is the solution of one additional scalar

algebraic equation for which a good initial guess is available.



Relaxation Runge-Kutta methods for inner-product norms are developed and

studied in

```

@article{ketcheson2019relaxation,

  title={Relaxation {R}unge-{K}utta Methods: {C}onservation and

         Stability for Inner-Product Norms},

  author={Ketcheson, David I},

  journal={SIAM Journal on Numerical Analysis},

  volume={57},

  number={6},

  pages={2850--2870},

  year={2019},

  publisher={Society for Industrial and Applied Mathematics},

  doi={10.1137/19M1263662},

  eprint={1905.09847},

  eprinttype={arxiv},

  eprintclass={math.NA}

}

```

Generalizations to general functionals with a particular emphasis on convex ones

are developed and studied in the following article, which is available on

[arXiv.org](https://arxiv.org/abs/1905.09129).

```

@article{ranocha2020relaxation,

  title={Relaxation {R}unge-{K}utta Methods: Fully-Discrete Explicit

         Entropy-Stable Schemes for the Compressible {E}uler and

         {N}avier-{S}tokes Equations},

  author={Ranocha, Hendrik and Sayyari, Mohammed and Dalcin, Lisandro and

          Parsani, Matteo and Ketcheson, David I},

  journal={SIAM Journal on Scientific Computing},

  volume={42},

  number={2},

  pages={A612--A638},

  year={2020},

  month={03},

  publisher={Society for Industrial and Applied Mathematics},

  doi={10.1137/19M1263480},

  eprint={1905.09129},

  eprinttype={arxiv},

  eprintclass={math.NA}

}

```

Implementations of relaxation Ruge-Kutta methods and numerical experiments

reported in that article can be found in this repository. If you find these

schemes useful, please cite the articles mentioned above. If you use the

implementations provided here, please cite this repository as

```

@misc{ranocha2019relaxationRepository,

  title={{ConvexRelaxationRungeKutta}. {R}elaxation {R}unge--{K}utta

         Methods for Convex Functionals},

  author={Ranocha, Hendrik and Ketcheson, David I.},

  year={2019},

  month={05},

  howpublished={\url{https://github.com/ranocha/ConvexRelaxationRungeKutta}},

  doi={10.5281/zenodo.3066518}

}

```

Specialised implementations of relaxation Runge-Kutta methods for inner-product

norms are available at https://github.com/ketch/RRK_rr.



## Disclaimer



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