Stability of step size control based on a posteriori error estimates

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

@online{ranocha2023stability,
  title={Stability of step size control based on a posteriori error estimates},
  author={Ranocha, Hendrik and Giesselmann, Jan},
  year={2023},
  month={07},
  doi={10.48550/arXiv.2307.12677},
  eprint={2307.12677},
  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{ranocha2023stabilityRepro,
  title={Reproducibility repository for
         "{S}tability of step size control based on a posteriori error estimates"},
  author={Ranocha, Hendrik and Giesselmann, Jan},
  year={2023},
  howpublished={\url{https://github.com/ranocha/2023_RK_error_estimate}},
  doi={10.5281/zenodo.8177157}
}

Abstract

A posteriori error estimates based on residuals can be used for reliable error control of numerical methods. Here, we consider them in the context of ordinary differential equations and Runge-Kutta methods. In particular, we take the approach of Dedner & Giesselmann (2016) and investigate it when used to select the time step size. We focus on step size control stability when combined with explicit Runge-Kutta methods and demonstrate that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable. We compare the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments.

Numerical experiments

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 v1.9.2.

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.

Authors

• Hendrik Ranocha (University of Hamburg, Germany)
• Jan Giesselmann (TU Darmstadt, Germany)

License

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.

Disclaimer

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