Abstract
For the solution of the one dimensional cubic nonlinear Schrödinger equation on the torus, we propose and analyze a fully discrete low-regularity integrator. The considered scheme is explicit. Its implementation relies on the fast Fourier transform with a complexity of \({\mathcal {O}}(N\log N)\) operations per time step, where N denotes the degrees of freedom in the spatial discretization. We prove that the new scheme provides an \({\mathcal {O}}(\tau ^{\frac{3}{2}\gamma -\frac{1}{2}-\varepsilon }+N^{-\gamma })\) error bound in \(L^2\) for any initial data in \(H^\gamma \), \(\frac{1}{2}<\gamma \le 1\), where \(\tau \) denotes the temporal step size. Numerical examples illustrate this convergence behavior.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Enquiries about data availability should be directed to the authors.
References
Besse, C., Bidégaray, B., Descombes, S.: Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40, 26–40 (2002)
Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3, 107–156 (1993)
Faou, E.: Geometric Numerical Integration and Schrödinger Equations. European Mathematical Society Publishing House, Zürich (2012)
Jahnke, T., Lubich, C.: Error bounds for exponential operator splittings. BIT 40, 735–744 (2000)
Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)
Knöller, M., Ostermann, A., Schratz, K.: A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data. SIAM J. Numer. Anal. 57, 1967–1986 (2019)
Li, B., Wu, Y.: A full discrete low-regularity integrator for the 1D period cubic nonlinear Schrödinger equation. Numer. Math. 149, 151–183 (2021)
Lubich, C.: On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77, 2141–2153 (2008)
Ostermann, A., Schratz, K.: Low regularity exponential-type integrators for semilinear Schrödinger equations. Found. Comput. Math. 18, 731–755 (2018)
Ostermann, A., Rousset, F., Schratz, K.: Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity. Found. Comput. Math. 21, 725–765 (2021)
Ostermann, A., Rousset, F., Schratz, K.: Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces, accepted for publication in J. Eur. Math. Soc.
Sanz-Serna, J.M.: Methods for the numerical solution of the nonlinear Schrödinger equation. Math. Comp. 43, 21–27 (1984)
Wu, Y., Yao, F.: A first-order Fourier integrator for the nonlinear Schrödinger equation on \({\mathbb{T}}\) without loss of regularity, accepted for publication in Math. Comp.
Funding
This research is supported by the NSFC key project under the Grant Number 11831003, and by NSFC under the Grant Number 11971356. The second author also acknowledges financial support by the China Scholarship Council.
Author information
Authors and Affiliations
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ostermann, A., Yao, F. A Fully Discrete Low-Regularity Integrator for the Nonlinear Schrödinger Equation. J Sci Comput 91, 9 (2022). https://doi.org/10.1007/s10915-022-01786-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-022-01786-y