GLIMPSE

This repository contains a collection of Matlab implementations of resonance-based low-regularity integrators and related methods that were studied over the course of the MSCA project GLIMPSE. The main routines can be found in src/resonance_based_schemes and examples illustrating the application of these methods as used in the below mentioned papers are provided in examples/. To run the examples please first run src/set_paths.m.

A good starting point to familiarise yourself with the code is given in examples/introductory_example/ where you can start by running convergence_experiments_nls.m followed by plotting.m.

Acknowledgements

If you use this code in an academic paper, please cite [1], [2], [3], [4]:

@misc{maierhofer2022symplectic,
     title={Bridging the gap: symplecticity and low regularity in {R}unge--{K}utta resonance-based schemes},
     author={Georg Maierhofer and Katharina Schratz},
     year={2022},
     eprint={2205.05024},
     archivePrefix={arXiv},
     primaryClass={math.NA}
  }

@misc{alamabronsard2023symmetric,
     title={Symmetric resonance based integrators and forest formulae},
     author={Yvonne Alama Bronsard and Yvain Bruned and Georg Maierhofer and Katharina Schratz},
     year={2023},
     eprint={2305.16737},
     archivePrefix={arXiv},
     primaryClass={math.NA}
  }

@misc{banica2022schroedingermaps,
     title={Numerical integration of {S}chr\"odinger maps via the {H}asimoto transform},
     author={Valeria Banica and Georg Maierhofer and Katharina Schratz},
     year={2022},
     eprint={2211.01282},
     archivePrefix={arXiv},
     primaryClass={math.NA}
  }

@misc{feng2023longtime,
     title={Long-time error bounds of low-regularity integrators for nonlinear {S}chr\"odinger equations},
     author={Yue Feng and Georg Maierhofer and Katharina Schratz},
     year={2023},
     eprint={2302.00383},
     archivePrefix={arXiv},
     primaryClass={math.NA}
  }

[1]	Maierhofer, G. and Schratz, K., “Bridging the gap: symplecticity and low regularity in Runge-Kutta resonance-based schemes”, 2022. arXiv.2205.05024.

[2]	Alama Bronsard, Y., Bruned, Y., Maierhofer, G. and Schratz, K., “Symmetric resonance based integrators and forest formulae”, 2023. arXiv.2305.16737.

[3]	Banica, V., Maierhofer, G. and Schratz, K., “Numerical integration of Schrödinger maps via the {H}asimoto transform”, 2022. arXiv.2211.01282, to appear in SIAM J. Numer. Anal.

[4]	Feng, Y., Maierhofer, G. and Schratz, K., “Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations”, 2023. arXiv.2302.00383, to appear in Math. Comput.