Skip to main content
SpringerLink
Log in

A fully discrete low-regularity integrator for the 1D periodic cubic nonlinear Schrödinger equation

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

A fully discrete and fully explicit low-regularity integrator is constructed for the one-dimensional periodic cubic nonlinear Schrödinger equation. The method can be implemented by using fast Fourier transform with \(O(N\ln N)\) operations at every time level, and is proved to have an \(L^2\)-norm error bound of \(O(\tau \sqrt{\ln (1/\tau )}+N^{-1})\) for \(H^1\) initial data, without requiring any CFL condition, where \(\tau \) and N denote the temporal stepsize and the degree of freedoms in the spatial discretisation, respectively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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)

    Article  MathSciNet  Google Scholar 

  2. 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)

    Article  MathSciNet  Google Scholar 

  3. Bourgain, J., Li, D.: On an endpoint Kato-Ponce inequality. Differ. Integral Equ. 27, 1037–1072 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. American Mathematical Society, Providence (2003)

    Google Scholar 

  5. Chu, E.: Discrete and Continuous Fourier Transforms Analysis, Applications and Fast Algorithms. CRC Press, New York (2008)

    Book  Google Scholar 

  6. Eilinghoff, J., Schnaubelt, R., Schratz, K.: Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation. J. Math. Anal. Appl. 442, 740–760 (2016)

    Article  MathSciNet  Google Scholar 

  7. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MathSciNet  Google Scholar 

  8. Hofmanováa, M., Schratz, K.: An exponential-type integrator for the KdV equation. Numer. Math. 136, 1117–1137 (2017)

    Article  MathSciNet  Google Scholar 

  9. Ignat, L.I.: A splitting method for the nonlinear Schrödinger equation. J. Differ. Equ. 250, 3022–3046 (2011)

    Article  Google Scholar 

  10. 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)

    Article  MathSciNet  Google Scholar 

  11. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)

    Article  MathSciNet  Google Scholar 

  12. Li, D.: On Kato-Ponce and fractional Leibniz. Rev. Mat. Iberoam. 35, 23–100 (2019)

    Article  MathSciNet  Google Scholar 

  13. Lubich, Ch.: On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77, 2141–2153 (2008)

    Article  Google Scholar 

  14. Ostermann, A., Rousset, F., Schratz, K.: Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity. Found. Comput. Math. (2020). https://doi.org/10.1007/s10208-020-09468-7

    Article  MATH  Google Scholar 

  15. Ostermann, A., Rousset, F., Schratz, K.: Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces. arXiv:2006.12785

  16. Ostermann, A., Schratz, K.: Low regularity exponential-type integrators for semilinear Schrödinger equations. Found. Comput. Math. 18, 731–755 (2018)

    Article  MathSciNet  Google Scholar 

  17. Ostermann, A., Su, C.: A Lawson-type exponential integrator for the Korteweg–de Vries equation. IMA J. Numer. Anal. (2020). https://doi.org/10.1093/imanum/drz030

    Article  MathSciNet  MATH  Google Scholar 

  18. Rousset, F., Schratz, K.: A general framework of low regularity integrators. arXiv:2010.01640

  19. Sanz-Serna, J.M.: Methods for the numerical solution of the nonlinear Schrödinger equation. Math. Comput. 43, 21–27 (1984)

    Article  Google Scholar 

  20. Schratz, K., Wang, Y., Zhao, X.: Low-regularity integrators for nonlinear Dirac equations. To appear in Mathematics of Computation (2020). arXiv:1906.09413

  21. Wang, J.: A new error analysis of Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. J. Sci. Comput. 60, 390–407 (2014)

    Article  MathSciNet  Google Scholar 

  22. Wu, Y., Yao, F.: A first-order Fourier integrator for the nonlinear Schrödinger equation on \({\mathbb{T}}\) without loss of regularity. Preprint, arXiv:2010.02672

  23. Wu, Y., Zhao, X.: Optimal convergence of a first order low-regularity integrator for the KdV equation. IMA Journal on Numerical Analysis. (2021). arXiv:1910.07367. https://doi.org/10.1093/imanum/drab054

  24. Wu, Y., Zhao, X.: Embedded exponential-type low-regularity integrators for KdV equation under rough data (2020). arXiv:2008.07053

  25. Wikipedia: https://en.wikipedia.org/wiki/Discrete_Fourier_transform

Download references

Funding

The work of B. Li is partially supported by Projects P0031035 ZZKQ and P0030125 ZZKK at The Hong Kong Polytechnic University. The work of Y. Wu is partially supported by NSFC 11771325 and 11571118.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

    Buyang Li

  2. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Yifei Wu

Authors
  1. Buyang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yifei Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Buyang Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, B., Wu, Y. A fully discrete low-regularity integrator for the 1D periodic cubic nonlinear Schrödinger equation. Numer. Math. 149, 151–183 (2021). https://doi.org/10.1007/s00211-021-01226-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-021-01226-3

Keywords

Mathematics Subject Classification

Search

Navigation