Traveling-wave solutions and structure-preserving numerical methods for a hyperbolic approximation of the Korteweg-de Vries equation

This repository contains information and code to reproduce the results presented in the article bibtex @article{biswas2025traveling, title={Traveling-wave solutions and structure-preserving numerical methods for a hyperbolic approximation of the {K}orteweg-de {V}ries equation}, author={Biswas, Abhijit and Ketcheson, David I. and Ranocha, Hendrik and Sch{\"u}tz, Jochen}, journal={Journal of Scientific Computing}, volume={103}, pages={90}, year={2025}, month={05}, doi={10.1007/s10915-025-02898-x}, eprint={2412.17117}, 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 bibtex @misc{biswas2024travelingRepro, title={Reproducibility repository for "{T}raveling-wave solutions and structure-preserving numerical methods for a hyperbolic approximation of the {K}orteweg-de {V}ries equation"}, author={Biswas, Abhijit and Ketcheson, David I. and Ranocha, Hendrik and Sch{\"u}tz, Jochen}, year={2024}, howpublished={\url{https://github.com/abhibsws/2024_kdvh_RR}}, doi={10.5281/zenodo.14423351} }

Abstract

We study the recently-proposed hyperbolic approximation of the Korteweg-de Vries equation (KdV). We show that this approximation, which we call KdVH, possesses a rich variety of solutions, including solitary wave solutions that approximate KdV solitons, as well as other solitary and periodic solutions that are related to higher-order water wave models, and may include singularities. We analyze a class of implicit-explicit Runge-Kutta time discretizations for KdVH that are asymptotic preserving, energy conserving, and can be applied to other hyperbolized systems. We also develop structure-preserving spatial discretizations based on summation-by-parts operators in space including finite difference, discontinuous Galerkin, and Fourier methods. We use the relaxation approach to make the fully discrete schemes energy-preserving. Numerical experiments demonstrate the effectiveness of these discretizations.

Numerical experiments

The numerical experiments use Python and Julia.

To run the Python code, you need Python 3, and the packages jupyter, numpy, scipy, and matplotlib. The Python code has been tested with the following versions, but other versions may work:

- Python 3.13
- Jupyter 1.1.1
- Numpy 2.1.3
- SciPy 1.14.1
- Matplotlib 3.9.2

We have used Julia version 1.10.7 for the experiments. The results can be reproduced by running all cells of the Jupyter notebook figures_manuscript.ipynb in order.

Disclaimer

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