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arxiv: 2503.22621 · v2 · submitted 2025-03-28 · 🧮 math.NA · cs.NA· math.AP

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Improved error estimates for low-regularity integrators using space-time bounds

Maximilian Ruff
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classification 🧮 math.NA cs.NAmath.AP
keywords estimatesintegratorsodingerschrwaveboundscaseconvergence
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We prove optimal convergence rates for certain low-regularity integrators applied to the one-dimensional periodic nonlinear Schr\"odinger and wave equations under the assumption of $H^1$ solutions. For the Schr\"odinger equation we analyze the exponential-type scheme proposed by Ostermann and Schratz in 2018, whereas in the wave case we treat the corrected Lie splitting proposed by Li, Schratz, and Zivcovich in 2023. We show that the integrators converge with their full order of one and two, respectively. In this situation only fractional convergence rates were previously known. The crucial ingredients in the proofs are known space-time bounds for the solutions to the corresponding linear problems. More precisely, in the Schr\"odinger case we use the $L^4$ Strichartz inequality, and for the wave equation a null form estimate. To our knowledge, this is the first time that a null form estimate is exploited in numerical analysis. We apply the estimates for continuous time, thus avoiding potential losses resulting from discrete-time estimates.

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Works this paper leans on

18 extracted references · 18 canonical work pages · cited by 2 Pith papers

  1. [1]

    Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I: Schrödinger equations

    J. Bourgain. “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I: Schrödinger equations”. In:Geom. Funct. Anal.3 (1993), pp. 107–156

    work page 1993

  2. [2]

    Resonance-based schemes for dispersive equations via decorated trees

    Y. Bruned and K. Schratz. “Resonance-based schemes for dispersive equations via decorated trees”. In:Forum Math. Pi10 (2022), pp. 1–76

    work page 2022

  3. [3]

    A new second-order low-regularity integrator for the cubic nonlinear Schrödinger equation

    J. Cao, B. Li, and Y. Lin. “A new second-order low-regularity integrator for the cubic nonlinear Schrödinger equation”. In:IMA J. Numer. Anal.44 (2024), pp. 1313–1345

    work page 2024

  4. [4]

    Y. Feng, G. Maierhofer, and C. Wang.Explicit symmetric low-regularity integrators for the nonlinear Schrödinger equation. arXiv preprint. 2024.doi: https://doi.org/ 10.48550/arXiv.2411.07720

    work page doi:10.48550/arxiv.2411.07720 2024

  5. [5]

    Error analysis of trigonometric integrators for semilinear Klein-Gordon equations

    L. Gauckler. “Error analysis of trigonometric integrators for semilinear Klein-Gordon equations”. In:SIAM J. Numer. Anal.53 (2015), pp. 1082–1106

    work page 2015

  6. [6]

    Exponential integrators

    M. Hochbruck and A. Ostermann. “Exponential integrators”. In:Acta Numerica19 (2010), pp. 209–286

    work page 2010

  7. [7]

    A splitting method for the nonlinear Schrödinger equation

    L. Ignat. “A splitting method for the nonlinear Schrödinger equation”. In:J. Differ- ential Equations250 (2011), pp. 3022–3046

    work page 2011

  8. [8]

    Space-time estimates for null forms and the local existence theorem

    S. Klainerman and M. Machedon. “Space-time estimates for null forms and the local existence theorem”. In:Comm. Pure Appl. Math.46 (1993), pp. 1221–1268

    work page 1993

  9. [9]

    A second-order low-regularity correction of Lie splitting for the semilinear Klein–Gordon equation

    B. Li, K. Schratz, and F. Zivcovich. “A second-order low-regularity correction of Lie splitting for the semilinear Klein–Gordon equation”. In:ESAIM Math. Model. Numer. Anal.57.2 (2023), pp. 899–919

    work page 2023

  10. [10]

    On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations

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

    work page 2008

  11. [11]

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

    A. Ostermann, F. Rousset, and K. Schratz. “Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity”. In:Found. Comput. Math.21 (2021), pp. 725–765

    work page 2021

  12. [12]

    Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces

    A. Ostermann, F. Rousset, and K. Schratz. “Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces”. In:J. Eur. Math. Soc.25.10 (2023), pp. 3913–3952. 14 REFERENCES

    work page 2023

  13. [13]

    Low regularity exponential-type integrators for semilinear Schrödinger equations

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

    work page 2018

  14. [14]

    A general framework of low regularity integrators

    F. Rousset and K. Schratz. “A general framework of low regularity integrators.” In: SIAM J. Numer. Anal.59.3 (2021), pp. 1735–1768

    work page 2021

  15. [15]

    Ruff.Error analysis of the Strang splitting for the 3D semilinear wave equation with finite-energy data

    M. Ruff.Error analysis of the Strang splitting for the 3D semilinear wave equation with finite-energy data. arXiv preprint. 2025.doi: https://doi.org/10.48550/ arXiv.2503.13126

    work page arXiv 2025

  16. [16]

    Error analysis of the Lie splitting for semilinear wave equations with finite-energy solutions

    M. Ruff and R. Schnaubelt. “Error analysis of the Lie splitting for semilinear wave equations with finite-energy solutions”. In:Discrete Contin. Dyn. Syst.(2025). Early Access. doi: 10.3934/dcds.2025009

    work page doi:10.3934/dcds.2025009 2025

  17. [17]

    Tao.Nonlinear Dispersive Equations: Local and Global Analysis

    T. Tao.Nonlinear Dispersive Equations: Local and Global Analysis. American Math- ematical Society, Providence, RI, 2006

    work page 2006

  18. [18]

    On Fourier coefficients and transforms of functions of two variables

    A. Zygmund. “On Fourier coefficients and transforms of functions of two variables”. In: Studia Math.50 (1974), pp. 189–201. Karlsruhe Institute of Technology, Department of Mathematics, Engler- straße 2, 76131 Karlsruhe, Germany Email address: maximilian.ruff@kit.edu

    work page 1974