Skip to main content
Log in

Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation based on recent time discretization and filtering techniques. For this new scheme, we perform a rigorous error analysis and establish better convergence rates at low regularity than known for classical schemes in the literature so far. In our error estimates, we combine the better local error properties of the new scheme with a stability analysis based on general discrete Strichartz-type estimates. The latter allow us to handle a much rougher class of solutions as the error analysis can be carried out directly at the level of \(L^2\) compared to classical results in dimension d, which are limited to higher-order (sufficiently smooth) Sobolev spaces \(H^s\) with \(s>d/2\). In particular, we are able to establish a global error estimate in \(L^2\) for \(H^1\) solutions which is roughly of order \(\tau ^{ {1\over 2} + { 5-d \over 12} }\) in dimension \(d \le 3\) (\(\tau \) denoting the time discretization parameter). This breaks the “natural order barrier” of \(\tau ^{1/2}\) for \(H^1\) solutions which holds for classical numerical schemes (even in combination with suitable filter functions).

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

Access this article

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. H. Bahouri, J.-Y. Chemin, R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Heidelberg, 2011.

    Book  Google Scholar 

  2. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part I: Schrödinger equations. Geom. Funct. Anal. 3:209–262 (1993).

  3. C. Besse, B. Bidégaray, S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40:26–40 (2002).

    Article  MathSciNet  Google Scholar 

  4. S. Baumstark, K. Schratz, Uniformly accurate oscillatory integrators for the Klein–Gordon–Zakharov system from low- to high-plasma frequency regimes. SIAM J. Numer. Anal. 57:429–457 (2019).

    Article  MathSciNet  Google Scholar 

  5. S. Baumstark, E. Faou, K. Schratz, Uniformly accurate oscillatory integrators for Klein–Gordon equations with asymptotic convergence to the classical NLS splitting. Math. Comp. 87:1227–1254 (2018).

    Article  MathSciNet  Google Scholar 

  6. N. Burq, P. Gérard, N. Tzvetkov. Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer. J. Math. 126:569–605 (2004).

    Article  MathSciNet  Google Scholar 

  7. B. Cano, A. González-Pachón, Exponential time integration of solitary waves of cubic Schrödinger equation. Appl. Numer. Math. 91:26–45 (2015).

    Article  MathSciNet  Google Scholar 

  8. T. Cazenave, Semilinear Schrödinger Equations. American Math. Soc., Providence RI, 2003.

    Book  Google Scholar 

  9. E. Celledoni, D. Cohen, B. Owren, Symmetric exponential integrators with an application to the cubic Schrödinger equation. Found. Comput. Math. 8:303–317 (2008).

    Article  MathSciNet  Google Scholar 

  10. W. Choi, Y. Koh, On the splitting method for the nonlinear Schrödinger equation with initial data in \(H^1\). Preprint (arxiv:1610.06028v3).

  11. D. Cohen, L. Gauckler, One-stage exponential integrators for nonlinear Schrödinger equations over long times. BIT 52:877–903 (2012).

    Article  MathSciNet  Google Scholar 

  12. G. Dujardin, Exponential Runge–Kutta methods for the Schrödinger equation. Appl. Numer. Math. 59:1839–1857 (2009).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. E. Faou, Geometric Numerical Integration and Schrödinger Equations. European Math. Soc. Publishing House, Zürich, 2012.

    Book  Google Scholar 

  15. L. Gauckler, C. Lubich, Nonlinear Schrödinger equations and their spectral semi-discretizations over long times. Found. Comput. Math. 20:141–169 (2010).

    Article  Google Scholar 

  16. L. Gauckler, Convergence of a split-step Hermite method for the Gross–Pitaevskii equation. IMA J. Numer. Anal. 31:396–415 (2011).

    Article  MathSciNet  Google Scholar 

  17. J. Ginibre, G. Velo. The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 2: 309–327 (1985).

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

    Article  MathSciNet  Google Scholar 

  19. M. Hofmanová, K. Schratz, An oscillatory integrator for the KdV equation, Numer. Math. 136:1117-1137 (2017).

    Article  MathSciNet  Google Scholar 

  20. L. I. Ignat, A splitting method for the nonlinear Schrödinger equation. J. Differential Equations 250:3022–3046 (2011).

    Article  MathSciNet  Google Scholar 

  21. L. Ignat, E. Zuazua, Dispersive properties of numerical schemes for nonlinear Schrödinger equations. In: L.M. Pardo, A. Pinkus, E. Suli, M.J. Todd (eds.), Foundations of Computational Mathematics Santander 2005. Cambridge Univ. Press, Cambridge, 2006, pp. 181–207.

    Chapter  Google Scholar 

  22. L. Ignat, E. Zuazua, Numerical dispersive schemes for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 47:1366–1390 (2009).

    Article  MathSciNet  Google Scholar 

  23. T. Jahnke, C. Lubich, Error bounds for exponential operator splittings. BIT, 40:735–744 (2000).

    Article  MathSciNet  Google Scholar 

  24. M. Keel, T. Tao. Endpoint Strichartz estimates. Amer. J. Math. 120:955–980 (1998).

    Article  MathSciNet  Google Scholar 

  25. P. Kunstmann, B. Li, C. Lubich, Runge–Kutta time discretization of nonlinear parabolic equations studied via discrete maximal parabolic regularity. Found. Comput. Math. 18:1109–1130 (2018).

    Article  MathSciNet  Google Scholar 

  26. F. Linares, G. Ponce, Introduction to Nonlinear Dispersive Equations. Second edition. Springer, New York, 2015.

    Book  Google Scholar 

  27. C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77:2141–2153 (2008).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. A. Stefanov, P. Kevrekidis, Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein–Gordon equations. Nonlinearity 18:1841–1857 (2005)

    Article  MathSciNet  Google Scholar 

  30. R. S. Strichartz. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44:705–714 (1977).

    Article  MathSciNet  Google Scholar 

  31. T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis. Amer. Math. Soc., Providence RI, 2006.

    Book  Google Scholar 

  32. M. Thalhammer, Convergence analysis of high-order time-splitting pseudo-spectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50:3231–3258 (2012).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

KS has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 850941).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Katharina Schratz.

Additional information

Communicated by Hans Munthe-Kaas.

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

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). https://doi.org/10.1007/s10208-020-09468-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-020-09468-7

Keywords

Mathematics Subject Classification

Navigation