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 and subsequent research projects. 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], [5]:

@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{banica2024schroedingermaps,

title={Numerical integration of {S}chr"odinger maps via the {H}asimoto transform}, author={Valeria Banica and Georg Maierhofer and Katharina Schratz}, journal={SIAM Journal on Numerical Analysis}, volume={62}, number={1}, pages={322--352}, year={2024}, publisher={SIAM}

}

@misc{feng2024longtime,

title={Long-time error bounds of low-regularity integrators for nonlinear {S}chr"odinger equations}, author={Yue Feng and Georg Maierhofer and Katharina Schratz}, journal={Mathematics of Computation}, volume={93}, number={348}, pages={1569--1598}, year={2024}

}

@misc{cui2025wongzakai,

title={A {W}ong--{Z}akai resonance-based integrator for the nonlinear {S}chr"odinger equation with white noise dispersion}, author={Jianbo Cui and Georg Maierhofer}, year={2025}, eprint={2503.19346}, 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, to appear in Math. Comp.

[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 Hasimoto transform”, SIAM J. Numer. Anal. 62(1), pp.322-352, 2024.

[4]	Feng, Y., Maierhofer, G. and Schratz, K., “Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations”, Math. Comp. 93(348), pp.1569-1598, 2024.

[5]	Cui, J. and Maierhofer, G., “A Wong--Zakai resonance-based integrator for the nonlinear Schrödinger equation with white noise dispersion”, 2025. arXiv:2503.19346.